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diff --git a/articles/discord.html b/articles/discord.html new file mode 100755 index 0000000..c3c4d33 --- /dev/null +++ b/articles/discord.html @@ -0,0 +1,220 @@ +<?xml version="1.0" encoding="utf-8"?> +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" +"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> +<head> +<!-- 2023-09-25 Mon 23:42 --> +<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> +<meta name="viewport" content="width=device-width, initial-scale=1" /> +<title>**Discord** : an internet cancer</title> +<meta name="author" content="Crystal" /> +<meta name="generator" content="Org Mode" /> +<link rel="stylesheet" type="text/css" href="../src/css/colors.css"/> +<link rel="stylesheet" type="text/css" href="../src/css/style.css"/> +</head> +<body> +<div id="content" class="content"> +<h1 class="title"><b><b>Discord</b></b> : an internet cancer</h1> + +<div id="outline-container-org7e1a396" class="outline-2"> +<h2 id="org7e1a396">Preamble</h2> +<div class="outline-text-2" id="text-org7e1a396"> +<p> +Before I start writing this article, I just want to clarify that I will NOT go over the technical aspect of <b><b>Discord</b></b> (such as the spyware and all) as it has been covered many times by other websites like <a href="https://spyware.neocities.org/articles/discord">This one !!!</a>, but basically, it’s exactly how you expect it to be, spying, selling data, monitoring open processes, terrible electron based app….etc +</p> + +<p> +I also wanted to make it clear that this is PURELY from my personal experience with <b><b>Discord</b></b> and the <b><b>Internet</b></b> as a whole, this isn’t an exact science and everybody can have an - although similar - different experience with the platform. Peace and love <3 +</p> +</div> +</div> + +<div id="outline-container-org54a193a" class="outline-2"> +<h2 id="org54a193a">Chapter one : Curiosity</h2> +<div class="outline-text-2" id="text-org54a193a"> +<p> +Picture yourself, it’s 2018/2019 and you are playing your favorite game, be it <b><b>Minecraft</b></b>, <b><b>League of Legends</b></b>, <b><b>CS:GO</b></b>, doesn’t matter. You start to play nicely, you make some friends, some enemies, typical gameplay. And then, one of them decides to take a step closer into your life, so they invite you to this cool new platform you have never heard about, <b><b>Discord</b></b>. Upon checking, you notice it’s a modern chat application for gamers…. “huh, must be nice” you might say. And then you are faced with a choice, you either create an account, or you don’t. +</p> + +<p> +If you decide to NOT make an account, you would be made fun of by that friend, or even get blamed for loses because “you weren’t in the voice chat”, you will be called a “Skype user”, probably miss out on events, giveaways, exclusive prizes and sometimes even get banned from that game server (Yeah, it happened on <b><b>Minecraft</b></b> before, failure to join the discord server actually rejects you from the Minecraft server). And even if we take a more…casual example, you found a cool piece of software online, but their guides and download links are all on <b><b>Discord</b></b>, or Discord exclusive, maybe you want to work with a team that solely use Discord. Basically you are missing out on a LOT. +</p> + +<p> +However, if you take the path of least resistance and decide to sign up, entering your e-mail address, password, birth date, and in the unfortunate case where you use a <b><b>VPN</b></b> or <b><b>TOR</b></b>, your phone number, pick a username and a profile picture. Congratulations, you now cursed yourself for the rest of your life. Remember that friend we imagined ? You play a few games together in VC, you crack jokes, you get to know each other… it’s all fun and games, but then, they send you a link you never saw before. Apparently it’s how you add people to groups in discord, they are called “Servers” +</p> + +<p> +> emkey.jpeg +</p> + +<p> +So you join that group they invited you to, it could be anything from a small friend group, to a large server with giveaways and events. But they all share one aspect, a hierarchy. +</p> +</div> +</div> + +<div id="outline-container-org3a0ea77" class="outline-2"> +<h2 id="org3a0ea77">Chapter two : The hierarchy</h2> +<div class="outline-text-2" id="text-org3a0ea77"> +<p> +Now this hierarchy is not inherently bad, but this is how <b><b>Discord</b></b> (and even the communities on Discord) keep you addicted to them. When you join, you start as a peasant, a pleb, a noobie even. You have an ugly color for your username, and no access to “channels”, only few ones with cooldowns so you don’t talk much. And then you look at the members list, and you see a beautiful rainbow, people divided into categories, or roles as they like to call them!! +</p> + +<p> +At the top you will most likely find the Owner(s) or their loved ones, with colors that inspire majesty and fear, like a dark purple, these are the masters of the server, they shape it the way they want and in the rare occasion where they join the plebs in their discussions, they are met with rounds of applause. +</p> + +<p> +Then just below them you have the Right hand(s) of the owners, generally veterans and original members, they have a bright, warm yet still majestic color to their role. These have the same power as the Owners, but are more active in the server and are actually hated by a lot of users, who unfairly got banned or kicked. +</p> + +<p> +Then just below them you have the Bots, non-human, non-sentient beings, most of the time with a silvery metallic color as their username which are there to execute the Owners commands, or just to annoy the plebs, like a tax collector after a long day of work <b>“Congratulations you just advanced to level 5”</b> +</p> + +<p> +Now, you have the Moderators, these have some power, have an aggressive color as their role and are generally the ones who try to control the Chaos in the server, although most of the time they fail due to a lack of permissions, or threats from the Owners themselves. Regardless, these are the ones with the biggest ego, and most importantly they are the ones causing server splits. +</p> + +<p> +If the aforementioned roles are considered black, and normal users are white, then Helpers are definitely the gray line separating them. They are peasants who strive to go a rank higher, to be a majesty, and have some power. But most of the time, they are there for an eternity, or they just go from there all the way up to a Right hand using some witchcraft….Oh and their color is almost always Green :D +</p> + +<p> +After that we have the Plebeians which are also separated in an attempt to control them easily. But I’m going to go through the list quickly otherwise this article will take ages : you have the supporters - people who helped the server financially or in other ways -, partners which are just owners of other servers who decided to become living billboards, you have event winners to remind the others of what they can achieve if they wait long enough, then we have the activity roles, the more you are active, the more you have rights over casual users, this is definitely not a way to keep you addicted to the group discussion ;3 +</p> + +<p> +And finally, you have the outcasts, warned or muted people are all the way down the list, and they have little to no rights or uses in the server, quite frankly if they left they would have a better time !!!! +</p> + + +<p> +Okay so, in all seriousness. This at first glance doesn’t seem like a bad idea, after all you need order and laws in an easily accessible group. But this is not the way it’s used, this hierarchy is constantly there to remind you of what you can be if you are liked enough by the owners, what you can achieve. And in most cases, you get access to exclusive channels, a secret club, creating a sense of scarcity and power. This is bad because it keeps people invested in discussions they honestly don’t want nor need to have, but they are obliged, otherwise they are just a “powerless normie”. +</p> + +<p> +Now what happens if a Mod or an Owner decides to abuse their power ? Well in this case, here are the different outcomes : +</p> + +<ul class="org-ul"> +<li>Either people get angry and start spamming and protesting (usually ends with a purge or a mass ban)</li> +<li>People just blame the victim because it’s more profitable</li> +<li>An other rogue mod decides to take control and destroy the server</li> +<li>Or if it’s a disagreement between Owners, the server undergoes a split</li> +</ul> + + +<p> +If you are paying attention, you would know that all of these will always end with the same ending : “Another server gets created”, and so this is how… +</p> +</div> +</div> + +<div id="outline-container-org2966135" class="outline-2"> +<h2 id="org2966135">Chapter three : <b><b>Discord</b></b> takes a once thriving community and splits it</h2> +<div class="outline-text-2" id="text-org2966135"> +<p> +Yes, there are always splits, and communities divide into multiple tiny sub-communities with their own opinions about useless matters. That is how you are kept invested. People love Drama, they love wars and they love picking sides. +</p> + +<p> +Let’s imagine together a simulation for what i mean, since i can’t draw or animate, use your imagination: +</p> +<ul class="org-ul"> +<li>You have a big red circle, its <b>Group A</b>, a pretty large tech community, with tiny dots inside with different colors, these are members and the colors represent their roles.</li> +<li>Once in a while, a fight happens in this group. Let’s say there is a fight between two normal users, a mod steps in and bans the user who is in the wrong. Not a big deal so far !</li> +<li>Now let’s imagine this scenario again, but with a different person, <b>Kevin</b>, a normal user in the group, but with a consistent presence, he is loved by a handful of people. Sadly he gets wrongly banned… wrong move for the mod, because now <b>Kevin</b> takes his supporters and make their own group, a blue one called <b>Group B</b></li> +<li><b>Group A</b>’s growth is starting to diminish because there is nothing to do there, and people are slowly moving to group <b>Group B</b> because of the overall aesthetic, in an attempt to win back their following, <b>Group A</b> decide to make events and reward their loyal followers. It kinda works but <b>Group B</b> is in a study growth</li> +<li>Oopsie, there is a minor disagreement between <b>Group A</b> owners, and it turns into a big war between them. which ends up with members taking sides and dividing the server in half, creating two new communities <b>Group C</b> and <b>Group D</b></li> +<li>And now, you know what is interesting ? is that all these 4 groups are not only semi-dead, but have the same users in them. AND SHARE THE SAME TOPIC</li> +</ul> + +<p> +So you basically killed a community, in 6 easy steps !!! And of course this will end up either by a mass deletion of these groups by a rogue Moderator, or a ban, or a screw up. So if the server contained important non-archived data. They are looong gone!!! +</p> +</div> +</div> + +<div id="outline-container-orga5ca424" class="outline-2"> +<h2 id="orga5ca424">Chapter four : <b><b>Discord</b></b> users are NOT your friends</h2> +<div class="outline-text-2" id="text-orga5ca424"> +<p> +<b><b>Discord</b></b> is made in a way that makes it easy to get attached to people, and also really hard to get rid of them, because you share the same servers, same friends, and the border between Private talk and Public talk is really blurred. Not to mention how hard, if not impossible it is to find someone who you met before but lost their contact. Because not only could they change their tag, but there is no way to search their username, and the servers can disappear from a minute to an other, or go private, or anything really !!! Now <a href="http://shystudios.us/blog/discord/discord.html">Shy actually talked about this issue on their article about Discord,</a> but here I’m making a different point, in their article they say that it’s hard to get rid of someone you know via Discord, which is absolutely true. But it’s also easy to lost contact with someone literally in a split second, even people you deem “close” to you, they just…disappear!! So for y’all thinking about dating on Discord, that’s a terrible idea !!!! +Imagine you’re in a <b><b>Discord</b></b> server, vibing with some awesome people, chatting about everything from the latest memes to the mysteries of the universe. You’ve become practically inseparable online pals, sharing inside jokes and bonding over your mutual hatred for pineapple on pizza. Life is grand, right? +</p> + +<p> +But, brace yourself for the plot twist: <b><b>Discord</b></b> friendships are like a box of chocolates - you never know what you’re going to get. People appear and disappear from servers faster than you can say “dank meme.” One day, your best Discord buddy is there, cracking jokes and sharing dog pics, and the next day, poof, they’re gone. Maybe they got bored, maybe real life called, or maybe they wandered into the Discord Bermuda Triangle. Who knows? +</p> + +<p> +Now, here’s the kicker: finding a lost <b><b>Discord</b></b> friend is like trying to find a grain of sand on a beach during a hurricane. You can’t just Google them, and even if you know their username, it’s about as useful as a chocolate teapot if they’ve changed it. Servers vanish, go private, or morph faster than a Pokémon in a battle. It’s like trying to capture smoke with a butterfly net. +</p> + +<p> +So, for those pondering the idea of <b><b>Discord</b></b> romance, think twice! While forming connections is a breeze, keeping tabs on those connections is like herding cats in a hurricane! +</p> +</div> +</div> + +<div id="outline-container-org41fdc9e" class="outline-2"> +<h2 id="org41fdc9e">Final Chapter : Login-walls</h2> +<div class="outline-text-2" id="text-org41fdc9e"> +<p> +People have made this point before and i will make it again, but locking important information behind a log-in page, with no way to find them using a Google search is stupid at best and manipulative at worst, because in this situation. Not only are you putting your faith on <b><b>Discord</b></b> servers to not fail one day, but on server Owners to not delete their work (and potentially rare unrecoverable work from other users). Not to mention that you actually need to be in that server to even know of the existence of these kind of resources. Regardless of how you see it, this is just putting valuable info in the hands of random people who could easily lock them behind a specific role that can be obtained either by paying, or by stroking their digital e-penis !!! +</p> + +<p> +You see, <b><b>Discord</b></b>, in all its infinite wisdom, believes it’s a brilliant idea to squirrel away precious knowledge behind a digital fortress that demands a username and password. It’s like saying, “Sure, I’ll share this life-changing information with you, but only if you can recall your umpteenth password!” +</p> + +<p> +Now, let’s break this down. First, you’re entrusting your prized data to <b><b>Discord</b></b> servers, which, let’s face it, are about as stable as a Jenga tower during an earthquake. One moment they’re there, and the next… poof! Gone with the wind. So much for your treasure trove of wisdom. +</p> + +<p> +And it gets even better. Server owners have the power to lock away valuable resources behind specific roles, which can be obtained through a combination of charm, flattery, or, heaven forbid, a cash transaction. It’s like saying, “Want to see the good stuff? Well, pony up or start groveling!” +</p> + +<p> +So, what’s the bottom line here? <b><b>Discord</b></b> has effectively become a modern-age Sphinx, guarding its secrets with a riddle of log-in screens. Your valuable info? In the hands of random folks who could decide to hoard it like misers guarding their gold or sell it to the highest bidder. It’s like a digital Wild West, and your information is the wild mustang everyone’s trying to wrangle. +</p> + +<p> +In conclusion, <b><b>Discord</b></b>’s penchant for login-walls is like locking away the Ark of the Covenant in a storage locker and hoping for the best. It’s the digital equivalent of hiding your keys in a haystack and hoping you can find them before the cows come home. +</p> +</div> +</div> +<div id="outline-container-org356f6e7" class="outline-2"> +<h2 id="org356f6e7">Conclusion</h2> +<div class="outline-text-2" id="text-org356f6e7"> +<p> +In this digital adventure, we’ve explored the mysterious realm of <b><b>Discord</b></b>, a platform that’s both a blessing and a curse. It’s a place where friendships blossom and vanish like shooting stars, where power dynamics create hierarchies that keep you hooked, and where valuable information is locked away like a dragon’s hoard. +</p> + +<p> +While <b><b>Discord</b></b> has its quirks and pitfalls, it’s essential to remember that it’s a reflection of the internet itself - a vast, ever-changing landscape filled with both wonder and peril. Your experience on Discord is uniquely yours, but it’s bound to be filled with surprises, friendships, and even the occasional drama. +</p> + +<p> +As we wrap up this exploration, it’s worth noting that <b><b>Discord</b></b>, like any digital space, is shaped by its users. The tales of adventures and misadventures, the rise and fall of servers, and the endless cycle of drama are all part of the grand tapestry of online life. +</p> + +<p> +So, if you find yourself lost in the labyrinthine corridors of <b><b>Discord</b></b> or stumble upon its peculiarities, remember, you’re not alone. Many have ventured before you, and many more will follow. The internet, after all, is an ever-evolving, enigmatic landscape where, as in life, every twist and turn holds the promise of a new adventure. +</p> + +<p> +If you ever have more anecdotes, insights, or questions to add to this digital saga, feel free to reach out. The story of <b><b>Discord</b></b> is far from over, and your voice could be the missing piece of the puzzle in this fascinating online journey. Until then, peace and love in your digital endeavors, and may your Discord adventures be filled with more joy than chaos! +</p> +</div> +</div> +</div> +<div id="postamble" class="status"> +<p class="author">Author: Crystal</p> +<p class="date">Created: 2023-09-25 Mon 23:42</p> +</div> +</body> +</html> \ No newline at end of file diff --git a/index.html b/index.html new file mode 100755 index 0000000..a71a9d9 --- /dev/null +++ b/index.html @@ -0,0 +1,167 @@ +<?xml version="1.0" encoding="utf-8"?> +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" +"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> +<head> +<!-- 2023-10-01 Sun 22:08 --> +<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> +<meta name="viewport" content="width=device-width, initial-scale=1" /> +<title>Crystal's Website 💜</title> +<meta name="author" content="Crystal" /> +<meta name="generator" content="Org Mode" /> +<link rel="stylesheet" type="text/css" href="src/css/colors.css"/> +<link rel="stylesheet" type="text/css" href="src/css/style.css"/> +</head> +<body> +<div id="content" class="content"> +<h1 class="title">Crystal’s Website 💜</h1> +<div id="outline-container-orgd8433bc" class="outline-2"> +<h2 id="orgd8433bc">Welcome to Crystal’s Cozy Nook!</h2> +<div class="outline-text-2" id="text-orgd8433bc"> +<p> +Hi there, <a href="./super_secret.html">adorable you!</a> +</p> + +<p> +I’m Crystal, and you’ve found my little online diary and treasure trove. 🌟 Here, I’m all about sharing the cutest and neatest things that brighten up my day. +</p> + +<p> +Imagine us sitting together, sipping hot cocoa, and having a delightful chat. It’s all about those heartwarming moments and the joy of simplicity. +</p> + +<p> +I’m over the moon that you’re here, adding even more sweetness to this corner! 💕 +</p> +</div> +</div> +<div id="outline-container-org9dc204a" class="outline-2"> +<h2 id="org9dc204a">Articles ( NEW !!!! )</h2> +<div class="outline-text-2" id="text-org9dc204a"> +<ul class="org-ul"> +<li><b><a href="./articles/discord.html">Discord : an internet cancer</a></b> <i>Sun Sep 10 15:25:22 2023</i></li> +</ul> +</div> +</div> +<div id="outline-container-org374ac9a" class="outline-2"> +<h2 id="org374ac9a">root@localhost $ whoami</h2> +<div class="outline-text-2" id="text-org374ac9a"> +<p> +My name is <b>Crystal</b>, I’m a <insert time since 09/12/2005> years old transfem from Algeria. Well I’m also pansexual, anarcho-communist and absurdist! I speak 4 languages (Kabyle/Tamazight Arabic French & English) if this info is of any use for you !! +</p> + +<p> +My current setup is : +</p> +<ul class="org-ul"> +<li>Primary OS: <b>OpenBSD -current</b></li> +<li>Text Editor: <b>Doom Emacs</b></li> +<li>Web Browser: <b>Ungoogled-chromium</b></li> +<li>Desktop Environment: <b>NsCDE</b></li> +<li>Shell: <b>Ksh</b></li> +<li>Secondary OS(For gaming): <b>OpenSUSE</b> with <b>KDE</b></li> +<li>Games I play: <b>Hollow Knight</b>, <b>Cult of the lamb</b>, <b>Dead Cells</b> & Rythm games(like <b>Phigros</b> and <b>A dance of fire and ice</b>)</li> +</ul> +<p> +I also host this website on the <a href="https://tilde.institute">https://tilde.institute</a> <b>pubNIX</b> which also runs <b>OpenBSD</b> +This might surprise you, but I also listen to music (A shocker, right?) though I mostly listen to <b>vaporwave</b> <b>glitchore</b> <b>weirdcore</b> <b>synthwave</b> and <b>dreamcore</b> +</p> + +<p> +If you want to contact me (which would be really surprising) contact me via <a href="mailto:crystal@danwin1210.de">mailto:crystal@danwin1210.de</a> +</p> + +<p> +My GNUPG (GPG) public key <a href="./src/txt/pubkey.asc">./src/txt/pubkey.asc</a> +</p> + +<p> +<a href="https://git.tilde.institute/crystal/www">This website is fully open-source with no licensing restrictions, check the source code and feel free to reuse everything!!!</a> +</p> +</div> +</div> +<div id="outline-container-org540901e" class="outline-2"> +<h2 id="org540901e">Sign my Guestbook (External website warning)</h2> +<div class="outline-text-2" id="text-org540901e"> +<p> +Want to leave a message, opinion, review or a salty insult ? Be sure to Sign my Guestbook then, it takes two seconds but it will mean the world to me !!! +</p> + + +<div id="org066036d" class="figure"> +<p><a href="https://crystaltilde.123guestbook.com/"><img src="./src/gifs/links/sign_my_guestbook-anim.gif" alt="sign_my_guestbook-anim.gif" /></a> +</p> +</div> +</div> +</div> +<div id="outline-container-orgb314855" class="outline-2"> +<h2 id="orgb314855">Blinkies</h2> +<div class="outline-text-2" id="text-orgb314855"> +<a href="http://validator.w3.org/check?uri=referer"><img + src="./src/gifs/blinkies/valid-xhtml10.png" alt="Valid XHTML 1.0 Strict" height="31" width="88" /></a> + <a href="https://jigsaw.w3.org/css-validator/check/referer"> + <img style="border:0;width:88px;height:31px" + src="./src/gifs/blinkies/vcss.gif" + alt="Valid CSS!" /> +</a> +<p> +<a href="https://nishi.boats/"><img src="./src/gifs/blinkies/nishiboats.jpg" alt="nishiboats.jpg" /></a> +<img src="./src/gifs/blinkies/girlsnow.png" alt="girlsnow.png" /> +<img src="./src/gifs/blinkies/cookiefree.gif" alt="cookiefree.gif" /> +<img src="./src/gifs/blinkies/transnow2.gif" alt="transnow2.gif" /> +<img src="./src/gifs/blinkies/gaywebring.gif" alt="gaywebring.gif" /> +<img src="./src/gifs/blinkies/tranarchy.gif" alt="tranarchy.gif" /> +<img src="./src/gifs/blinkies/button-torrents.gif" alt="button-torrents.gif" /> +<img src="./src/gifs/blinkies/tyg.gif" alt="tyg.gif" /> +<img src="./src/gifs/blinkies/fuck-google.gif" alt="fuck-google.gif" /> +<img src="./src/gifs/blinkies/fuck_facebook.gif" alt="fuck_facebook.gif" /> +<img src="./src/gifs/blinkies/graphics_by_gimp.gif" alt="graphics_by_gimp.gif" /> +<img src="./src/gifs/blinkies/learn_html.gif" alt="learn_html.gif" /> +<img src="./src/gifs/blinkies/leave-twitter.gif" alt="leave-twitter.gif" /> +<img src="./src/gifs/blinkies/stop_microsoft.gif" alt="stop_microsoft.gif" /> +<img src="./src/gifs/blinkies/web-pi.png" alt="web-pi.png" /> +<img src="./src/gifs/blinkies/piracy.gif" alt="piracy.gif" /> +<img src="./src/gifs/blinkies/best_viewed_with_eyes.gif" alt="best_viewed_with_eyes.gif" /> +<a href="https://spyware.neocities.org/articles/discord"><img src="./src/gifs/blinkies/discord-no-way-2.gif" alt="discord-no-way-2.gif" /></a> +<a href="https://yesterweb.org/no-to-web3/"><img src="./src/gifs/blinkies/roly-saynotoweb3.gif" alt="roly-saynotoweb3.gif" /></a> +</p> + + +<p> +<a href="https://openbsd.org/"><img src="./src/gifs/blinkies/openbsd.png" alt="openbsd.png" /></a> +<a href="https://partysepe13.neocities.org/"><img src="./src/gifs/blinkies/partysepe.png" alt="partysepe.png" /></a> +</p> +</div> +<div id="outline-container-org59abd1a" class="outline-3"> +<h3 id="org59abd1a">My banner</h3> +<div class="outline-text-3" id="text-org59abd1a"> +<p> +If you enjoyed my website, you could link me on your personal website using this banner. If you don’t want to, then no pressure 💜 I still love you and I hope that this small shrine of mine will impress you in the future!!! +</p> + + +<div id="org42e22aa" class="figure"> +<p><img src="./src/gifs/crystal-tilde.gif" alt="crystal-tilde.gif" /> +</p> +</div> +</div> +</div> +</div> +<div id="outline-container-org45ca9be" class="outline-2"> +<h2 id="org45ca9be"><a href="https://crystal.tilde.institute/links.html">Webrings & Links (JAVASCRIPT WARNING)!!</a></h2> +</div> +<div id="outline-container-org4e92f4e" class="outline-2"> +<h2 id="org4e92f4e">Misc :</h2> +<div class="outline-text-2" id="text-org4e92f4e"> +<ol class="org-ol"> +<li><b><a href="./uni_notes/">My University notes</a></b></li> +</ol> +</div> +</div> +</div> +<div id="postamble" class="status"> +<p class="author">Author: Crystal</p> +<p class="date">Created: 2023-10-01 Sun 22:08</p> +</div> +</body> +</html> \ No newline at end of file diff --git a/links.html b/links.html new file mode 100755 index 0000000..65ceae9 --- /dev/null +++ b/links.html @@ -0,0 +1,95 @@ +<?xml version="1.0" encoding="utf-8"?> +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" +"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> +<head> +<!-- 2023-09-15 Fri 20:23 --> +<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> +<meta name="viewport" content="width=device-width, initial-scale=1" /> +<title>Webrings and Links (JavaScript ahead) 💜</title> +<meta name="author" content="Crystal" /> +<meta name="generator" content="Org Mode" /> +<link rel="stylesheet" type="text/css" href="src/css/colors.css"/> +<link rel="stylesheet" type="text/css" href="src/css/style.css"/> +</head> +<body> +<div id="content" class="content"> +<h1 class="title">Webrings and Links (JavaScript ahead) 💜</h1> +<div id="outline-container-org0f47f24" class="outline-2"> +<h2 id="org0f47f24">Webrings & Links</h2> +<div class="outline-text-2" id="text-org0f47f24"> +<p> +<b>This site is a proud member of the geekring! Check some other geeky websites here!</b> +</p> + +<p> +<a href="http://geekring.net/site/302/previous">Previous site</a> – <a href="http://geekring.net/site/301/random">Random Site</a> – <a href="http://geekring.net/site/301/next">Next Site</a> +</p> + +<p> +<b>Do you long for a simpler time, when America was Online and the only person you could Ask was Jeeves? Hotline Webring is bringing that time back, with Webrings! <i>This website is part of the Hotline Webring</i></b> +</p> + +<p> +<a href="https://hotlinewebring.club/crystal/previous">Previous site</a> – <a href="https://hotlinewebring.club/crystal/next">Next site</a> +</p> +<iframe id="bucket-webring" style="width: 100%; height: 3rem; border: none;" src="https://webring.bucketfish.me/embed.html?name=crystal"></iframe> + +<link rel="stylesheet" +href="https://teethinvitro.neocities.org/webring/linuxring/script/onionring.css"> +<div id="transring"> +<script type="text/javascript" src="https://transring.neocities.org/onionring-variables.js"></script> +<script type="text/javascript" src="https://transring.neocities.org/onionring-widget.js"></script> +</div> + +<div id='linuxring'> +<script type="text/javascript" src="https://teethinvitro.neocities.org/webring/linuxring/script/onionring-variables.js"></script> +<script type="text/javascript" src="https://teethinvitro.neocities.org/webring/linuxring/script/onionring-widget.js"></script> +</div> +<table> +<tr> +<td><a href="https://webri.ng/webring/ladiesofthelinks/previous?via=https%3A%2F%2Fcrystal.tilde.institute"><img src="src/gifs/links/ladiesofthelinks/ladiesofthelink1.gif"></a></td> +<td><a href="https://ladiesofthe.link/"><img src="src/gifs/links/ladiesofthelinks/ladiesofthelink.gif"></a></td> +<td><a href="https://webri.ng/webring/ladiesofthelinks/next?via=https%3A%2F%2Fcrystal.tilde.institute"><img src="/src/gifs/links/ladiesofthelinks/ladiesofthelink2.gif"></a></td> +</tr> +</table> +</div> + +<div id="outline-container-org614fbc6" class="outline-3"> +<h3 id="org614fbc6">Lainchan Webring</h3> +<div class="outline-text-3" id="text-org614fbc6"> +<p> +Lainring is a decentralized <a href="https://indieweb.org/webring">webring</a> created by the users of <a href="https://www.lainchan.org">Lainchan</a>, an anonymous image board. If you want to be added, go to the <a href="https://lainchan.org/%CE%A9/res/70358.html">Lainchan thread</a> and post your website there, together with a 240x60 button image. +</p> + +<div id="lainring">... Loading, please wait ...</div> +<script> +document.addEventListener("DOMContentLoaded", function(event) { + /* Try to retrieve the json file */ + fetch('src/json/lainring.json').then(res => res.json()).then((data) => { + let out = ''; + /* For each element in the JSON, build an anchor-image DOM structure */ + data.items.forEach(element => { + /* This string is split in multiple lines for readability */ + out += '<a title="' + element.title + '" ' + + 'href="' + element.url + '">' + + '<img src="src/gifs/lainchan/' + element.img + '" alt="' + element.title + '" /></a>'; + }); + /* Inject the DOM structure into the element with the id 'lainring' */ + document.getElementById('lainring').innerHTML = out; + }).catch(err => { + /* throw an error */ + throw err + }); +}); +</script> +</div> +</div> +</div> +</div> +<div id="postamble" class="status"> +<p class="author">Author: Crystal</p> +<p class="date">Created: 2023-09-15 Fri 20:23</p> +</div> +</body> +</html> \ No newline at end of file diff --git a/src/css/colors.css b/src/css/colors.css new file mode 100755 index 0000000..fed702b --- /dev/null +++ b/src/css/colors.css @@ -0,0 +1,11 @@ +:root { + --color0: #06050B; + --color1: #a686f7; + --color2: #9C6793; + --color3: #DA629B; + --color4: #C35CB8; + --color5: #A998A0; + --color6: #D1A5A8; + --color7: #e2d8d6; + --color8: #9e9795; +} diff --git a/src/css/style.css b/src/css/style.css new file mode 100755 index 0000000..d43f734 --- /dev/null +++ b/src/css/style.css @@ -0,0 +1,115 @@ +@font-face { + font-family: basiic; + src: url(../fonts/basiic.ttf); +} +@font-face { + font-family: 'Special Elite'; + font-style: normal; + font-weight: 400; + font-display: swap; + src: url(../fonts/Special_Elite.woff2) format('woff2'); + unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+0304, U+0308, U+0329, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD; +} +html { + font-family: "Special Elite"; + background-color: var(--color0); + max-width: 70em; + margin: auto; +} + +h1 { + color: var(--color1); + text-align: center; + margin-bottom: .2em; +} + +h2 { + color: var(--color2); +} + +p { + color: var(--color1); +} +table { + color: var(--color1); +} +a { + color: var(--color4); +} + +li { + color: var(--color8); +} + +h3 { + color: var(--color6); +} + +h4 { + color: var(--color7); +} + +b { + color: var(--color3); +} + +div.outline-2 { + border: 4px solid var(--color5); + padding: 10px; + margin: 10px; +} + +img { + max-width: 100%; + height: auto; +} + +.figure p { + text-align: center; +} + +body::before { + content: ""; + display: block; + position: fixed; + top: 0; + left: 0; + bottom: 0; + right: 0; + background: linear-gradient(rgba(20, 20, 20, 0) 66%, #DA629B0B 33%); + background-size: 100% 9px; + z-index: 9999; + pointer-events: none; + animation: scanlines-anim 1.0s linear infinite; +} + +@keyframes scanlines-anim { + 0% { + background-position: 0px 0px; + } + 100% { + background-position: 0px 15px; + } +} + +@media only screen and (orientation: portrait) { + body::before { + background-size: 100% 4px; + } +} +@keyframes rotating { + from { + + transform: rotate(0deg); + } + to { + transform: rotate(360deg); + } +} + +.smurf { +animation: rotating 2s linear infinite; + max-width: 40%; + height: auto; + +} diff --git a/src/fonts/Special_Elite.woff2 b/src/fonts/Special_Elite.woff2 new file mode 100755 index 0000000..569a27e --- /dev/null +++ b/src/fonts/Special_Elite.woff2 Binary files differdiff --git a/src/fonts/basiic.ttf b/src/fonts/basiic.ttf new file mode 100755 index 0000000..b0e6d09 --- /dev/null +++ b/src/fonts/basiic.ttf Binary files differdiff --git a/src/gifs/blinkies/best_viewed_with_eyes.gif b/src/gifs/blinkies/best_viewed_with_eyes.gif new file mode 100755 index 0000000..6dfeb64 --- /dev/null +++ b/src/gifs/blinkies/best_viewed_with_eyes.gif Binary files differdiff --git a/src/gifs/blinkies/button-torrents.gif 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","img":"grafo.zone.gif","title":"Grafo.zone"},{"url":"https://mkultravict.im/","img":"mkultravict.im.jpg","title":"MkUltraVictim"}]} + diff --git a/src/media/welive.mp3 b/src/media/welive.mp3 new file mode 100755 index 0000000..6ce97b2 --- /dev/null +++ b/src/media/welive.mp3 Binary files differdiff --git a/src/org/articles/discord.org b/src/org/articles/discord.org new file mode 100755 index 0000000..a6d6c63 --- /dev/null +++ b/src/org/articles/discord.org @@ -0,0 +1,107 @@ +#+TITLE: **Discord** : an internet cancer +#+AUTHOR: Crystal +#+OPTIONS: ^:{} +#+OPTIONS: num:nil +#+EXPORT_FILE_NAME: ../../../articles/discord.html +#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../src/css/colors.css"/> +#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../src/css/style.css"/> +#+OPTIONS: html-style:nil +#+OPTIONS: toc:nil + +* Preamble + +Before I start writing this article, I just want to clarify that I will NOT go over the technical aspect of **Discord** (such as the spyware and all) as it has been covered many times by other websites like [[https://spyware.neocities.org/articles/discord][This one !!!]], but basically, it's exactly how you expect it to be, spying, selling data, monitoring open processes, terrible electron based app....etc + +I also wanted to make it clear that this is PURELY from my personal experience with **Discord** and the **Internet** as a whole, this isn't an exact science and everybody can have an - although similar - different experience with the platform. Peace and love <3 + +* Chapter one : Curiosity + +Picture yourself, it's 2018/2019 and you are playing your favorite game, be it **Minecraft**, **League of Legends**, **CS:GO**, doesn't matter. You start to play nicely, you make some friends, some enemies, typical gameplay. And then, one of them decides to take a step closer into your life, so they invite you to this cool new platform you have never heard about, **Discord**. Upon checking, you notice it's a modern chat application for gamers.... "huh, must be nice" you might say. And then you are faced with a choice, you either create an account, or you don't. + +If you decide to NOT make an account, you would be made fun of by that friend, or even get blamed for loses because "you weren't in the voice chat", you will be called a "Skype user", probably miss out on events, giveaways, exclusive prizes and sometimes even get banned from that game server (Yeah, it happened on **Minecraft** before, failure to join the discord server actually rejects you from the Minecraft server). And even if we take a more...casual example, you found a cool piece of software online, but their guides and download links are all on **Discord**, or Discord exclusive, maybe you want to work with a team that solely use Discord. Basically you are missing out on a LOT. + +However, if you take the path of least resistance and decide to sign up, entering your e-mail address, password, birth date, and in the unfortunate case where you use a **VPN** or **TOR**, your phone number, pick a username and a profile picture. Congratulations, you now cursed yourself for the rest of your life. Remember that friend we imagined ? You play a few games together in VC, you crack jokes, you get to know each other... it's all fun and games, but then, they send you a link you never saw before. Apparently it's how you add people to groups in discord, they are called "Servers" + +> emkey.jpeg + +So you join that group they invited you to, it could be anything from a small friend group, to a large server with giveaways and events. But they all share one aspect, a hierarchy. + +* Chapter two : The hierarchy + +Now this hierarchy is not inherently bad, but this is how **Discord** (and even the communities on Discord) keep you addicted to them. When you join, you start as a peasant, a pleb, a noobie even. You have an ugly color for your username, and no access to "channels", only few ones with cooldowns so you don't talk much. And then you look at the members list, and you see a beautiful rainbow, people divided into categories, or roles as they like to call them!! + +At the top you will most likely find the Owner(s) or their loved ones, with colors that inspire majesty and fear, like a dark purple, these are the masters of the server, they shape it the way they want and in the rare occasion where they join the plebs in their discussions, they are met with rounds of applause. + +Then just below them you have the Right hand(s) of the owners, generally veterans and original members, they have a bright, warm yet still majestic color to their role. These have the same power as the Owners, but are more active in the server and are actually hated by a lot of users, who unfairly got banned or kicked. + +Then just below them you have the Bots, non-human, non-sentient beings, most of the time with a silvery metallic color as their username which are there to execute the Owners commands, or just to annoy the plebs, like a tax collector after a long day of work *"Congratulations you just advanced to level 5"* + +Now, you have the Moderators, these have some power, have an aggressive color as their role and are generally the ones who try to control the Chaos in the server, although most of the time they fail due to a lack of permissions, or threats from the Owners themselves. Regardless, these are the ones with the biggest ego, and most importantly they are the ones causing server splits. + +If the aforementioned roles are considered black, and normal users are white, then Helpers are definitely the gray line separating them. They are peasants who strive to go a rank higher, to be a majesty, and have some power. But most of the time, they are there for an eternity, or they just go from there all the way up to a Right hand using some witchcraft....Oh and their color is almost always Green :D + +After that we have the Plebeians which are also separated in an attempt to control them easily. But I'm going to go through the list quickly otherwise this article will take ages : you have the supporters - people who helped the server financially or in other ways -, partners which are just owners of other servers who decided to become living billboards, you have event winners to remind the others of what they can achieve if they wait long enough, then we have the activity roles, the more you are active, the more you have rights over casual users, this is definitely not a way to keep you addicted to the group discussion ;3 + +And finally, you have the outcasts, warned or muted people are all the way down the list, and they have little to no rights or uses in the server, quite frankly if they left they would have a better time !!!! + + +Okay so, in all seriousness. This at first glance doesn't seem like a bad idea, after all you need order and laws in an easily accessible group. But this is not the way it's used, this hierarchy is constantly there to remind you of what you can be if you are liked enough by the owners, what you can achieve. And in most cases, you get access to exclusive channels, a secret club, creating a sense of scarcity and power. This is bad because it keeps people invested in discussions they honestly don't want nor need to have, but they are obliged, otherwise they are just a "powerless normie". + +Now what happens if a Mod or an Owner decides to abuse their power ? Well in this case, here are the different outcomes : + +- Either people get angry and start spamming and protesting (usually ends with a purge or a mass ban) +- People just blame the victim because it's more profitable +- An other rogue mod decides to take control and destroy the server +- Or if it's a disagreement between Owners, the server undergoes a split + + + If you are paying attention, you would know that all of these will always end with the same ending : "Another server gets created", and so this is how... + +* Chapter three : **Discord** takes a once thriving community and splits it + +Yes, there are always splits, and communities divide into multiple tiny sub-communities with their own opinions about useless matters. That is how you are kept invested. People love Drama, they love wars and they love picking sides. + +Let's imagine together a simulation for what i mean, since i can't draw or animate, use your imagination: +- You have a big red circle, its *Group A*, a pretty large tech community, with tiny dots inside with different colors, these are members and the colors represent their roles. +- Once in a while, a fight happens in this group. Let's say there is a fight between two normal users, a mod steps in and bans the user who is in the wrong. Not a big deal so far ! +- Now let's imagine this scenario again, but with a different person, *Kevin*, a normal user in the group, but with a consistent presence, he is loved by a handful of people. Sadly he gets wrongly banned... wrong move for the mod, because now *Kevin* takes his supporters and make their own group, a blue one called *Group B* +- *Group A*'s growth is starting to diminish because there is nothing to do there, and people are slowly moving to group *Group B* because of the overall aesthetic, in an attempt to win back their following, *Group A* decide to make events and reward their loyal followers. It kinda works but *Group B* is in a study growth +- Oopsie, there is a minor disagreement between *Group A* owners, and it turns into a big war between them. which ends up with members taking sides and dividing the server in half, creating two new communities *Group C* and *Group D* +- And now, you know what is interesting ? is that all these 4 groups are not only semi-dead, but have the same users in them. AND SHARE THE SAME TOPIC + +So you basically killed a community, in 6 easy steps !!! And of course this will end up either by a mass deletion of these groups by a rogue Moderator, or a ban, or a screw up. So if the server contained important non-archived data. They are looong gone!!! + +* Chapter four : **Discord** users are NOT your friends + +**Discord** is made in a way that makes it easy to get attached to people, and also really hard to get rid of them, because you share the same servers, same friends, and the border between Private talk and Public talk is really blurred. Not to mention how hard, if not impossible it is to find someone who you met before but lost their contact. Because not only could they change their tag, but there is no way to search their username, and the servers can disappear from a minute to an other, or go private, or anything really !!! Now [[http://shystudios.us/blog/discord/discord.html][Shy actually talked about this issue on their article about Discord,]] but here I'm making a different point, in their article they say that it's hard to get rid of someone you know via Discord, which is absolutely true. But it's also easy to lost contact with someone literally in a split second, even people you deem "close" to you, they just...disappear!! So for y'all thinking about dating on Discord, that's a terrible idea !!!! +Imagine you're in a **Discord** server, vibing with some awesome people, chatting about everything from the latest memes to the mysteries of the universe. You've become practically inseparable online pals, sharing inside jokes and bonding over your mutual hatred for pineapple on pizza. Life is grand, right? + +But, brace yourself for the plot twist: **Discord** friendships are like a box of chocolates - you never know what you're going to get. People appear and disappear from servers faster than you can say "dank meme." One day, your best Discord buddy is there, cracking jokes and sharing dog pics, and the next day, poof, they're gone. Maybe they got bored, maybe real life called, or maybe they wandered into the Discord Bermuda Triangle. Who knows? + +Now, here's the kicker: finding a lost **Discord** friend is like trying to find a grain of sand on a beach during a hurricane. You can't just Google them, and even if you know their username, it's about as useful as a chocolate teapot if they've changed it. Servers vanish, go private, or morph faster than a Pokémon in a battle. It's like trying to capture smoke with a butterfly net. + +So, for those pondering the idea of **Discord** romance, think twice! While forming connections is a breeze, keeping tabs on those connections is like herding cats in a hurricane! + +* Final Chapter : Login-walls + +People have made this point before and i will make it again, but locking important information behind a log-in page, with no way to find them using a Google search is stupid at best and manipulative at worst, because in this situation. Not only are you putting your faith on **Discord** servers to not fail one day, but on server Owners to not delete their work (and potentially rare unrecoverable work from other users). Not to mention that you actually need to be in that server to even know of the existence of these kind of resources. Regardless of how you see it, this is just putting valuable info in the hands of random people who could easily lock them behind a specific role that can be obtained either by paying, or by stroking their digital e-penis !!! + +You see, **Discord**, in all its infinite wisdom, believes it's a brilliant idea to squirrel away precious knowledge behind a digital fortress that demands a username and password. It's like saying, "Sure, I'll share this life-changing information with you, but only if you can recall your umpteenth password!" + +Now, let's break this down. First, you're entrusting your prized data to **Discord** servers, which, let's face it, are about as stable as a Jenga tower during an earthquake. One moment they're there, and the next... poof! Gone with the wind. So much for your treasure trove of wisdom. + +And it gets even better. Server owners have the power to lock away valuable resources behind specific roles, which can be obtained through a combination of charm, flattery, or, heaven forbid, a cash transaction. It's like saying, "Want to see the good stuff? Well, pony up or start groveling!" + +So, what's the bottom line here? **Discord** has effectively become a modern-age Sphinx, guarding its secrets with a riddle of log-in screens. Your valuable info? In the hands of random folks who could decide to hoard it like misers guarding their gold or sell it to the highest bidder. It's like a digital Wild West, and your information is the wild mustang everyone's trying to wrangle. + +In conclusion, **Discord**'s penchant for login-walls is like locking away the Ark of the Covenant in a storage locker and hoping for the best. It's the digital equivalent of hiding your keys in a haystack and hoping you can find them before the cows come home. +* Conclusion +In this digital adventure, we've explored the mysterious realm of **Discord**, a platform that's both a blessing and a curse. It's a place where friendships blossom and vanish like shooting stars, where power dynamics create hierarchies that keep you hooked, and where valuable information is locked away like a dragon's hoard. + +While **Discord** has its quirks and pitfalls, it's essential to remember that it's a reflection of the internet itself - a vast, ever-changing landscape filled with both wonder and peril. Your experience on Discord is uniquely yours, but it's bound to be filled with surprises, friendships, and even the occasional drama. + +As we wrap up this exploration, it's worth noting that **Discord**, like any digital space, is shaped by its users. The tales of adventures and misadventures, the rise and fall of servers, and the endless cycle of drama are all part of the grand tapestry of online life. + +So, if you find yourself lost in the labyrinthine corridors of **Discord** or stumble upon its peculiarities, remember, you're not alone. Many have ventured before you, and many more will follow. The internet, after all, is an ever-evolving, enigmatic landscape where, as in life, every twist and turn holds the promise of a new adventure. + +If you ever have more anecdotes, insights, or questions to add to this digital saga, feel free to reach out. The story of **Discord** is far from over, and your voice could be the missing piece of the puzzle in this fascinating online journey. Until then, peace and love in your digital endeavors, and may your Discord adventures be filled with more joy than chaos! diff --git a/src/org/class_notes.org b/src/org/class_notes.org new file mode 100755 index 0000000..fe1d9a1 --- /dev/null +++ b/src/org/class_notes.org @@ -0,0 +1,97 @@ +#+title: Class notes + + +* class 1 /sun sep 17 13:04:43 2023/ +** Les Reseaux locaux (LAN) +Cable 10 mbps ou moins = Ethernet +Cable 100 mbps = Fast Ethernet +Cable 1000 mbps ou plus = Giga Ethernet + +** MAC Address : +First three Bytes : vendor +Last three Bytes : Unique ID +(In Hexadecimal) + +TODO: CSMA/ CA CD + +** RJ-45 connecteurs +**** Straight +**** Crossover (croisé) +**** Rollover (console) + +** IPV4 classes : +- Classe A : 0 - 127 /8 + - 0.0.0.0/0 : route par default + - 127.0.0.0/8 : bouclage +- Classe B : 128 - 191 /16 + - 169.254.0.0/16 : APIPA +- Classe C : 192 - 223 /24 +- Classe D : 224 - 239 + - Multicast +- Classe E : 240 - 255 + +** Adresses privées: +- Classe A : 10.0.0.0 - 10.255.255.255 /8 +- Classe B : 172.16.0.0 - 172.31.255.255 /12 +- Classe C : 192.168.0.0 - 192.168.255.255 /16 + + +8 7 6 5 4 3 2 1 +128 64 32 16 8 4 2 1 +256 128 64 32 16 8 4 2 +/24 /25 /26 /27 /28 /29 /30 /31 + +500 +150 +60 +30 +20 +15 +** 500 hosts : + +Addresse: 10.0.0.0 /23 + +500 - 256 = 244 + +First IP : 10.0.0.1 /23 +Last IP : 10.0.1.254 /23 +@B IP : 10.0.1.255 /23 +Mask : 255.255.254.0 + +** 150 hosts : + +Addresse: 10.0.2.0 /24 +First IP: 10.0.2.1 /24 +Last IP: 10.0.2.254 /24 +@B IP: 10.0.2.255 /24 +Mask: 255.255.255.0 + +** 60 hosts : +Addresse: 10.0.3.0 /26 +First IP: 10.0.3.1 /26 +Last IP: 10.0.3.63 /26 +@B IP: 10.0.3.64 /26 +Mask: 255.255.255.191 + +** 30 hosts: + Addresse: 10.0.3.63/8 + First IP: 10.0.3.64/8 + Last IP: 10.0.3.96/8 + @B IP: 10.0.3.97/8 + Mask: 255.255.255.158/11 + +** 20 +Addresse: 10.0.3.97/8 +First IP: 10.0.3.98/8 +Last IP: 10.0.3.128/8 +@B IP: 10.0.3.129/8 +Mask: 255.255.255.126/11 +** 15 +Addresse: 10.0.3.129/8 +First IP: 10.0.3.130/8 +Last IP: 10.0.3.144/8 +@B IP: 10.0.3.145/8 +Mask: 255.255.255.110/12 + +* Class 2Sun /Sep 24 09:09:28 2023/ +** CCTV (finally) diff --git a/src/org/index.org b/src/org/index.org new file mode 100755 index 0000000..305f519 --- /dev/null +++ b/src/org/index.org @@ -0,0 +1,86 @@ +#+TITLE: Crystal's Website 💜 +#+AUTHOR: Crystal +#+OPTIONS: ^:{} +#+OPTIONS: num:nil +#+EXPORT_FILE_NAME: ../../index.html +#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="src/css/colors.css"/> +#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="src/css/style.css"/> + +#+OPTIONS: html-style:nil +#+OPTIONS: toc:nil +* Welcome to Crystal's Cozy Nook! +Hi there, [[./super_secret.html][adorable you!]] + +I'm Crystal, and you've found my little online diary and treasure trove. 🌟 Here, I'm all about sharing the cutest and neatest things that brighten up my day. + +Imagine us sitting together, sipping hot cocoa, and having a delightful chat. It's all about those heartwarming moments and the joy of simplicity. + +I'm over the moon that you're here, adding even more sweetness to this corner! 💕 + +* Articles ( NEW !!!! ) +- *[[./articles/discord.html][Discord : an internet cancer]]* /Sun Sep 10 15:25:22 2023/ +* root@localhost $ whoami +My name is *Crystal*, I'm a <insert time since 09/12/2005> years old transfem from Algeria. Well I'm also pansexual, anarcho-communist and absurdist! I speak 4 languages (Kabyle/Tamazight Arabic French & English) if this info is of any use for you !! + +My current setup is : +- Primary OS: *OpenBSD -current* +- Text Editor: *Doom Emacs* +- Web Browser: *Ungoogled-chromium* +- Desktop Environment: *NsCDE* +- Shell: *Ksh* +- Secondary OS(For gaming): *OpenSUSE* with *KDE* +- Games I play: *Hollow Knight*, *Cult of the lamb*, *Dead Cells* & Rythm games(like *Phigros* and *A dance of fire and ice*) +I also host this website on the https://tilde.institute *pubNIX* which also runs *OpenBSD* +This might surprise you, but I also listen to music (A shocker, right?) though I mostly listen to *vaporwave* *glitchore* *weirdcore* *synthwave* and *dreamcore* + +If you want to contact me (which would be really surprising) contact me via mailto:crystal@danwin1210.de + +My GNUPG (GPG) public key [[./src/txt/pubkey.asc]] + +[[https://git.tilde.institute/crystal/www][This website is fully open-source with no licensing restrictions, check the source code and feel free to reuse everything!!!]] + +* Sign my Guestbook (External website warning) +Want to leave a message, opinion, review or a salty insult ? Be sure to Sign my Guestbook then, it takes two seconds but it will mean the world to me !!! + +[[https://crystaltilde.123guestbook.com/][file:./src/gifs/links/sign_my_guestbook-anim.gif]] +* Blinkies +#+BEGIN_EXPORT html + <a href="http://validator.w3.org/check?uri=referer"><img + src="./src/gifs/blinkies/valid-xhtml10.png" alt="Valid XHTML 1.0 Strict" height="31" width="88" /></a> + <a href="https://jigsaw.w3.org/css-validator/check/referer"> + <img style="border:0;width:88px;height:31px" + src="./src/gifs/blinkies/vcss.gif" + alt="Valid CSS!" /> + </a> +#+END_EXPORT +[[https://nishi.boats/][file:./src/gifs/blinkies/nishiboats.jpg]] +[[./src/gifs/blinkies/girlsnow.png]] +[[./src/gifs/blinkies/cookiefree.gif]] +[[./src/gifs/blinkies/transnow2.gif]] +[[./src/gifs/blinkies/gaywebring.gif]] +[[./src/gifs/blinkies/tranarchy.gif]] +[[./src/gifs/blinkies/button-torrents.gif]] +[[./src/gifs/blinkies/tyg.gif]] +[[./src/gifs/blinkies/fuck-google.gif]] +[[./src/gifs/blinkies/fuck_facebook.gif]] +[[./src/gifs/blinkies/graphics_by_gimp.gif]] +[[./src/gifs/blinkies/learn_html.gif]] +[[./src/gifs/blinkies/leave-twitter.gif]] +[[./src/gifs/blinkies/stop_microsoft.gif]] +[[./src/gifs/blinkies/web-pi.png]] +[[./src/gifs/blinkies/piracy.gif]] +[[./src/gifs/blinkies/best_viewed_with_eyes.gif]] +[[https://spyware.neocities.org/articles/discord][file:./src/gifs/blinkies/discord-no-way-2.gif]] +[[https://yesterweb.org/no-to-web3/][file:./src/gifs/blinkies/roly-saynotoweb3.gif]] + + +[[https://openbsd.org/][file:./src/gifs/blinkies/openbsd.png]] +[[https://partysepe13.neocities.org/][file:./src/gifs/blinkies/partysepe.png]] + +** My banner +If you enjoyed my website, you could link me on your personal website using this banner. If you don't want to, then no pressure 💜 I still love you and I hope that this small shrine of mine will impress you in the future!!! + +[[./src/gifs/crystal-tilde.gif]] +* [[https://crystal.tilde.institute/links.html][Webrings & Links (JAVASCRIPT WARNING)!!]] +* Misc : +1. *[[./uni_notes/][My University notes]]* diff --git a/src/org/links.org b/src/org/links.org new file mode 100755 index 0000000..610b667 --- /dev/null +++ b/src/org/links.org @@ -0,0 +1,71 @@ +#+TITLE: Webrings and Links (JavaScript ahead) 💜 +#+AUTHOR: Crystal +#+OPTIONS: ^:{} +#+OPTIONS: num:nil +#+EXPORT_FILE_NAME: ../../links.html +#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="src/css/colors.css"/> +#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="src/css/style.css"/> +#+OPTIONS: html-style:nil +#+OPTIONS: toc:nil +* Webrings & Links + +*This site is a proud member of the geekring! Check some other geeky websites here!* + +[[http://geekring.net/site/302/previous][Previous site]] -- [[http://geekring.net/site/301/random][Random Site]] -- [[http://geekring.net/site/301/next][Next Site]] + +*Do you long for a simpler time, when America was Online and the only person you could Ask was Jeeves? Hotline Webring is bringing that time back, with Webrings! /This website is part of the Hotline Webring/* + +[[https://hotlinewebring.club/crystal/previous][Previous site]] -- [[https://hotlinewebring.club/crystal/next][Next site]] +#+BEGIN_EXPORT html +<iframe id="bucket-webring" style="width: 100%; height: 3rem; border: none;" src="https://webring.bucketfish.me/embed.html?name=crystal"></iframe> + +<link rel="stylesheet" +href="https://teethinvitro.neocities.org/webring/linuxring/script/onionring.css"> +<div id="transring"> +<script type="text/javascript" src="https://transring.neocities.org/onionring-variables.js"></script> +<script type="text/javascript" src="https://transring.neocities.org/onionring-widget.js"></script> +</div> + +<div id='linuxring'> +<script type="text/javascript" src="https://teethinvitro.neocities.org/webring/linuxring/script/onionring-variables.js"></script> +<script type="text/javascript" src="https://teethinvitro.neocities.org/webring/linuxring/script/onionring-widget.js"></script> +</div> + +#+END_EXPORT +#+BEGIN_EXPORT html +<table> +<tr> +<td><a href="https://webri.ng/webring/ladiesofthelinks/previous?via=https%3A%2F%2Fcrystal.tilde.institute"><img src="src/gifs/links/ladiesofthelinks/ladiesofthelink1.gif"></a></td> +<td><a href="https://ladiesofthe.link/"><img src="src/gifs/links/ladiesofthelinks/ladiesofthelink.gif"></a></td> +<td><a href="https://webri.ng/webring/ladiesofthelinks/next?via=https%3A%2F%2Fcrystal.tilde.institute"><img src="/src/gifs/links/ladiesofthelinks/ladiesofthelink2.gif"></a></td> +</tr> +</table> +#+END_EXPORT + +** Lainchan Webring + +Lainring is a decentralized [[https://indieweb.org/webring][webring]] created by the users of [[https://www.lainchan.org][Lainchan]], an anonymous image board. If you want to be added, go to the [[https://lainchan.org/%CE%A9/res/70358.html][Lainchan thread]] and post your website there, together with a 240x60 button image. + +#+BEGIN_EXPORT html +<div id="lainring">... Loading, please wait ...</div> +<script> +document.addEventListener("DOMContentLoaded", function(event) { + /* Try to retrieve the json file */ + fetch('src/json/lainring.json').then(res => res.json()).then((data) => { + let out = ''; + /* For each element in the JSON, build an anchor-image DOM structure */ + data.items.forEach(element => { + /* This string is split in multiple lines for readability */ + out += '<a title="' + element.title + '" ' + + 'href="' + element.url + '">' + + '<img src="src/gifs/lainchan/' + element.img + '" alt="' + element.title + '" /></a>'; + }); + /* Inject the DOM structure into the element with the id 'lainring' */ + document.getElementById('lainring').innerHTML = out; + }).catch(err => { + /* throw an error */ + throw err + }); +}); +</script> +#+END_EXPORT diff --git a/src/org/super_secret.org b/src/org/super_secret.org new file mode 100755 index 0000000..b892d01 --- /dev/null +++ b/src/org/super_secret.org @@ -0,0 +1,25 @@ +#+TITLE: SUPER SECRET WEBSITE +#+AUTHOR: Crystal +#+OPTIONS: ^:{} +#+OPTIONS: num:nil +#+EXPORT_FILE_NAME: ../../super_secret.html +#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="src/css/colors.css"/> +#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="src/css/style.css"/> +#+OPTIONS: html-style:nil +#+OPTIONS: toc:nil +#+BEGIN_EXPORT HTML +<div class="smurf"> +#+END_EXPORT + +[[./src/gifs/smurf.jpg]] +#+BEGIN_EXPORT HTML +</div> +#+END_EXPORT + + +#+BEGIN_EXPORT HTML +<audio controls autoplay loop> + <source src="./src/media/welive.mp3" type="audio/mpeg"> + Your browser does not support the audio element. +</audio> +#+END_EXPORT diff --git a/src/org/uni_notes/algebra1.org b/src/org/uni_notes/algebra1.org new file mode 100755 index 0000000..4afc389 --- /dev/null +++ b/src/org/uni_notes/algebra1.org @@ -0,0 +1,496 @@ +#+title: Algebra 1 +#+AUTHOR: Crystal +#+OPTIONS: ^:{} +#+OPTIONS: num:nil +#+EXPORT_FILE_NAME: ../../../uni_notes/algebra.html +#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../src/css/colors.css"/> +#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../src/css/style.css"/> +#+OPTIONS: html-style:nil +#+OPTIONS: toc:nil + +* Contenu de la Matiére +** Rappels et compléments (11H) +- Logique mathématique et méthodes du raisonnement mathématique +- Ensembles et Relations +- Applications + +** Structures Algébriques (11H) +- Groupes et morphisme de groupes +- Anneaux et morphisme d'anneaux +- Les corps + +** Polynômes et fractions rationnelles +- Notion du polynôme à une indéterminée á coefficients dans un anneau +- Opérations Algébriques sur les polynômes +- Arithmétique dans l'anneau des polynômes +- Polynôme dérivé et formule de Taylor +- Notion de racine d'un polynôme +- Notion de Fraction rationelle á une indéterminée +- Décomposition des fractions rationelles en éléments simples + +* Premier cours : Logique mathématique et méthodes du raisonnement mathématique /Sep 25/ : + +Let *P* *Q* and *R* be propositions which can either be *True* or *False*. And let's also give the value *1* to each *True* proposition and *0* to each false one. + +/Ex:/ +- *5 ≥ 2* is a proposition, a correct one !!! +- *The webmaster is a girl* is also a proposition, which is also correct. +- *x is always bigger than 5* is *not* a proposition, because we CAN'T determine if it's correct or not as *x* changes. +...etc + +In order to avoid repetition, and rewriting the proposition over and over, we just assign a capital letter to them such as *P Q* or *R*. + +So now we could write : +*Let the proposition P be 5 ≥ 2, we notice that P is always True, therefor its validity is 1* + +We also have the opposite of *P*, which is *not(P)* but for simplicity we use *P̅* (A P with a bar on top, in case it doesn't load for you), now let's go back to the previous example: + +*Since we know that the proposition P is true, we can conclude that P̅ is false. As P and P̅ can NOT be true at the same time. It's like saying 5 is greater and also lesser than 2...doesn't make sense, does it ?* + +Now let's say we have two propositions, and we want to test the validity of their disjunction..... Okay what is this "disjunction" ? *Great Question Billy !!!* A disjunction is true if either propositions are true + +Ex: +*Let proposition P be "The webmaster is asleep", and Q be "The reader loves pufferfishes". The disjunction of these two propositions can have 4 different values showed in this Table of truth (such a badass name):* + +| P | Q | Disjunction | +|---+---+-------------| +| 1 | 1 | 1 | +| 1 | 0 | 1 | +| 0 | 1 | 1 | +| 0 | 0 | 0 | + +/What the hell is this ?/ +The first colomn is equivalent to saying : "The webmaster is asleep AND The reader loves pufferfishes" +The second one means : "The webmaster is asleep AND The reader DOESN'T love pufferfishes (if you are in this case, then *I HATE YOU*)" +The third one... /zzzzzzz/ + +You got the idea !!! +And since we are talking about a disjunction here, *one of the propositions* need to be true in order for this disjunction to be true. + +You may be wondering.... Crystal, can't we write a disjunction in magical math symbols ? And to this I respond with a big *YES*. A disjunction is symbolized by a *∨* . So the disjunction between proposition *P & Q* can be written this way : *P ∨ Q* + +What if, we want to test whether or not two propositions are true AT THE SAME TIME ? Long story short, we can, it's called a conjunction, same concept, as before, only this time the symbol is *P ∧ Q*, and is only true if *P* and *Q* are true. So we get a Table like this : + +| P | Q | P ∨ Q | P ∧ Q | +|---+---+-------+-------| +| 1 | 1 | 1 | 1 | +| 1 | 0 | 1 | 0 | +| 0 | 1 | 1 | 0 | +| 0 | 0 | 0 | 0 | + +*Always remember: 1 means true and 0 means false* + +There are two more basics to cover here before going to some properties, the first one is implication symbolized by the double arrow *⇒* + +Implication is kinda hard for my little brain to explain, so I will just say what it means: + +*If P implies Q, this means that either Q, or the opposite of P are correct* + +or in math terms + +*P ⇒ Q translates to P̅ ∨ Q* +Let's illustrate : + +| P | Q | P̅ | Q̅ | P ∨ Q | P ∧ Q | P ⇒ Q (P̅ ∨ Q) | +|---+---+---+---+-------+-------+---------------| +| 1 | 1 | 0 | 0 | 1 | 1 | 1 | +| 1 | 0 | 0 | 1 | 1 | 0 | 0 | +| 0 | 1 | 1 | 0 | 1 | 0 | 1 | +| 0 | 0 | 1 | 1 | 0 | 0 | 1 | + +*If you look clearly, there is only one case where an implication is false. therefor you just need to find it, and blindly say that the others are correct. A rule of thumb is that: "A correct never implies a false", or "If a 1 tries to imply a 0, the implication is a 0"* + +Aight, a last one and we are done!!! Equivalence, which is fairly easy, symbolized by a *⇔* symbol. + +A proposition is equivalent to another only when both of them have *the same value of truth* AKA: both true or both false. a little table will help demonstrate what i mean. + +| P | Q | P̅ | Q̅ | P ∨ Q | P ∧ Q | P ⇒ Q (P̅ ∨ Q) | P ⇔ Q | +|---+---+---+---+-------+-------+---------------+-------| +| 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | +| 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | +| 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | +| 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | + +/Note: P implying Q is equivalent to P̅ implying Q̅, or: (P ⇒ Q) ⇔ (P̅ ⇒ Q̅)/ + +** Properties: +*** *Absorption*: +(P ∨ P) ⇔ P + +(P ∧ P) ⇔ P + +*** *Commutativity*: +(P ∧ Q) ⇔ (Q ∧ P) + +(P ∨ Q) ⇔ (Q ∨ P) + +*** *Associativity*: +P ∧ (Q ∧ R) ⇔ (P ∧ Q) ∧ R + +P ∨ (Q ∨ R) ⇔ (P ∨ Q) ∨ R + +*** *Distributivity*: +P ∧ (Q ∨ R) ⇔ (P ∧ Q) ∨ (P ∧ R) + +P ∨ (Q ∧ R) ⇔ (P ∨ Q) ∧ (P ∨ R) + +*** *Neutral element*: +/We define proposition *T* to be always *true* and *F* to be always *false*/ + +P ∧ T ⇔ P + +P ∨ F ⇔ P +*** *Negation of a conjunction & a disjunction*: +Now we won't use bars here because my lazy ass doesn't know how, so instead I will use not()!!! + +not(*P ∧ Q*) ⇔ P̅ ∨ Q̅ + +not(*P ∨ Q*) ⇔ P̅ ∧ Q̅ + +*A rule I really like to use here is: Break and Invert. Basically you break the bar into the three characters of the propositions, so you get not(P) not(∧ or ∨) /NOT AN ACTUAL MATH WRITING. DONT USE IT ANYWHERE ELSE OTHER THAN YOUR BRAIN/ and not(Q)* + +*** *Transitivity*: +[(P ⇒ Q) (Q ⇒ R)] ⇔ P ⇒ R + +*** *Contraposition*: +(P ⇒ Q) ⇔ (Q̅ ⇒ P̅) + +*** God only knows what this property is called: +/If/ + +(P ⇒ Q) is true + +and + +(Q̅ ⇒ Q) is true + +then + +Q is always true + +** Some exercices I found online : + +*** USTHB 2022/2023 Section B : + +**** Exercice 1: Démontrer les équivalences suivantes: +1. (P ⇒ Q) ⇔ (Q̅ ⇒ P̅) + + Basically we are asked to prove contraposition, so here we have ( P ⇒ Q ) which is equivalent to P̅ ∨ Q *By definition : (P ⇒ Q) ⇔ (P̅ ∨ Q)* + + + So we end up with : *(P̅ ∨ Q) ⇔ (Q̅ ⇒ P̅)*, now we just do the same with the second part of the contraposition. *(Q̅ ⇒ P̅) ⇔ (Q ∨ P̅)* therefor : + + + *(Q ∨ P̅) ⇔ (P̅ ∨ Q)*, which is true because of commutativity + +2. not(P ⇒ Q) ⇔ P ∧ Q̅ + + +Okaaaay so, let's first get rid of the implication, because I don't like it : *not(P̅ ∨ Q)* + + +Now that we got rid of it, we can negate the whole disjunction *not(P̅ ∨ Q) ⇔ (P ∧ Q̅)*. Which is the equivalence we needed to prove + +3. P ⇒ (Q ∧ R) ⇔ (P ⇒ Q) ∧ (P ⇒ R) + + One might be tempted to replace P with P̅ to get rid of the implication...sadly this isnt it. All we have to do here is resort to *Distributivity*, because yeah, we can distribute an implication across a {con/dis}junction + +4. P ∧ (Q ∨ R) ⇔ (P ∧ Q) ∨ (P ∧ R) + + Literally the same as above 🩷 + + +**** Exercice 2: Dire si les propositions suivantes sont vraies ou fausses, et les nier: + +1. ∀x ∈ ℝ ,∃y ∈ ℝ*+, tels que e^x = y + + For each x from the set of Real numbers, there exists a number y from the set of non-zero positive Real numbers that satisfies the equation : e^x = y + + +"The function f(x)=e^x is always positive and non-null", the very definition of an exponential function !!!! + + +*So the proposition is true* + + +2. ∃x ∈ ℝ, tels que x^2 < x < x^3 + + +We just need to find a value that satisifies this condition...thankfully its easy.... + +x² < x < x³ , we divide the three terms by x so we get : + + +x < 1 < x² , or : + + +*x < 1* ; *1 < x²* ⇔ *x < 1* ; *1 < x* /We square root both sides/ + + +We end up with a contradiction, therefor its wrong + + +3. ∀x ∈ ℝ, ∃y ∈ ℝ tels que y = 3x - 8 + + +I dont really understand this one, so let me translate it "For any value of x from the set of Real numbers, 3x - 8 is a Real number".... i mean....yeah, we are substracting a Real number from an other real number... + +*Since substraction is an Internal composition law in ℝ, therefor all results of a substraction between two Real numbers is...Real* + +4. ∃x ∈ ℕ, ∀y ∈ ℕ, x > y ⇒ x + y < 8 + + "There exists a number x from the set of Natural numbers such as for all values of y from the set of Natural numbers, x > y implies x + y < 8" + + +Let's get rid of the implication : + +∃x ∈ ℕ, ∀y ∈ ℕ, (y > x) ∨ (x + y < 8) /There exists a number x from the set of Natural numbers such as for all values of y from the set of Natural numbers y > x OR x + y < 8/ + +This proposition is true, because there exists a value of x that satisfies this condition, it's *all numbers under 8* let's take 3 as an example: + + +*x = 3 , if y > 3 then the first condition is true ; if y < 3 then the second one is true* + + +Meaning that the two propositions CAN NOT BE WRONG TOGETHER, either one is wrong, or the other + + +y > x + + +*y - x > 0* + + +y + x < 8 + + +*y < 8 - x* /This one is always true for all values of x below 8, since we are working in the set ℕ/ + + +5. ∀x ∈ ℝ, x² ≥ 1 ⇔ x ≥ 1 + + ....This is getting stupid. of course it's true it's part of the definition of the power of 2 + + +* 2éme cours /Oct 2/ + +** Quantifiers + +A propriety P can depend on a parameter x + + +∀ is the universal quantifier which stands for "For any value of..." + + +∃ is the existential quantifier which stands for "There exists at least one..." + + +***** Example +P(x) : x+1≥0 + +P(X) is True or False depending on the values of x + + +*** Proprieties +**** Propriety Number 1: +The negation of the universal quantifier is the existential quantifier, and vice-versa : + +- not(∀x ∈ E , P(x)) ⇔ ∃ x ∈ E, not(P(x)) +- not(∃x ∈ E , P(x)) ⇔ ∀ x ∈ E, not(P(x)) + +***** Example: +∀ x ≥ 1 x² > 5 ⇔ ∃ x ≥ 1 x² < 5 +**** Propriety Number 2: + +*∀x ∈ E, [P(x) ∧ Q(x)] ⇔ [∀ x ∈ E, P(x)] ∧ [∀ x ∈ E, Q(x)]* + + +The propriety "For any value of x from a set E , P(x) and Q(x)" is equivalent to "For any value of x from a set E, P(x) AND for any value of x from a set E, Q(x)" +***** Example : +P(x) : sqrt(x) > 0 ; Q(x) : x ≥ 1 + + +∀x ∈ ℝ*+, [sqrt(x) > 0 , x ≥ 1] ⇔ [∀x ∈ R*+, sqrt(x) > 0] ∧ [∀x ∈ R*+, x ≥ 1] + + +*Which is true* +**** Propriety Number 3: + +*∃ x ∈ E, [P(x) ∧ Q(x)] /⇒/ [∃ x ∈ E, P(x)] ∧ [∃ x ∈ E, Q(x)]* + + +/Here its an implication and not an equivalence/ + +***** Example of why it's NOT an equivalence : +P(x) : x > 5 ; Q(x) : x < 5 + + +Of course there is no value of x such as its inferior and superior to 5 at the same time, so obviously the proposition is false. However, the two propositions separated are correct on their own, because there is a value of x such as its superior to 5, and there is also a value of x such as its inferior to 5. This is why it's an implication and NOT AN EQUIVALENCE!!! +**** Propriety Number 4: + +*[∀ x ∈ E, P(x)] ∨ [∀ x ∈ E, Q(x)] /⇒/ ∀x ∈ E, [P(x) ∨ Q(x)]* + + +/Same here, implication and NOT en equivalence/ + + +** Multi-parameter proprieties : + +A propriety P can depend on two or more parameters, for convenience we call them x,y,z...etc + +***** Example : +P(x,y): x+y > 0 + + +P(0,1) is a True proposition + + +P(-2,-1) is a False one + +***** WARNING : + +∀x ∈ E, ∃y ∈ F , P(x,y) + + +∃y ∈ F, ∀x ∈ E , P(x,y) + + +Are different because in the first one y depends on x, while in the second one, it doesn't +****** Example : +∀ x ∈ ℕ , ∃ y ∈ ℕ y > x ------ True + + +∃ y ∈ ℕ , ∀ x ∈ ℕ y > x ------ False + +**** Proprieties : +1. not(∀x ∈ E ,∃y ∈ F P(x,y)) ⇔ ∃x ∈ E, ∀y ∈ F not(P(x,y)) +2. not(∃x ∈ E ,∀y ∈ F P(x,y)) ⇔ ∀x ∈ E, ∃y ∈ F not(P(x,y)) + +** Methods of mathematical reasoning : +*** Direct reasoning : + +To show that an implication P ⇒ Q is true, we suppose that P is true and we show that Q is true + +**** Example: +Let a,b be two Real numbers, we have to prove that *a² + b² = 1 ⇒ |a + b| ≤ 2* + + +We suppose that a²+b² = 1 and we prove that |a + b| ≤ 2 + + +a²+b²=1 ⇒ b² = 1 - a² ; a² = 1 - b² + + +a²+b²=1 ⇒ 1 - a² ≥ 0 ; 1 - b² ≥ 0 + + +a²+b²=1 ⇒ a² ≤ 1 ; b² ≤ 1 + + +a²+b²=1 ⇒ -1 ≤ a ≤ 1 ; -1 ≤ b ≤ 1 + + +a²+b²=1 ⇒ -2 ≤ a + b ≤ 2 + + +a²+b²=1 ⇒ |a + b| ≤ 2 *Which is what we wanted to prove, therefor the implication is correct* + +*** Reasoning by the Absurd: +To prove that a proposition is True, we suppose that it's False and we must come to a contradiction + + +And to prove that an implication P ⇒ Q is true using the reasoning by the absurd, we suppose that P ∧ not(Q) is true, and then we come to a contradiction as well +**** Example: +Prove that this proposition is correct using the reasoning by the absurd : ∀x ∈ ℝ* , sqrt(1+x²) ≠ 1 + x²/2 + + +We assume that ∃ x ℝ* , sqrt(1+x²) = 1 + x²/2 + + +sqrt(1+x²) = 1 + x²/2 ; 1 + x² = (1+x²/2)² ; 1 + x² = 1 + x^4/4 + x² ; x^(4)/4 = 0 ... Which contradicts with our proposition, since x = 4 and we are working on the ℝ* set + + +*** Reasoning by contraposition: +If an implication P ⇒ Q is too hard to prove, we just have to prove not(Q) ⇒ not(P) is true !!! or in other words that both not(P) and not(Q) are true + + +*** Reasoning by counter example: +To prove that a proposition ∀x ∈ E, P(x) is false, all we have to do is find a single value of x from E such as not(P(x)) is true +* 3eme Cours : /Oct 9/ +*** Reasoning by recurrence : +P is a propriety dependent of *n ∈ ℕ*. If for n0 ∈ ℕ P(n0) is true, and if for n ≥ n0 (P(n) ⇒ P(n+1)) is true. Then P(n) is true for n ≥ n0 + +**** Example: +Let's prove that ∀ n ≥ 1 , (n,k=1)Σk = [n(n+1)]/2 + + +P(n) : (n,k=1)Σk = [n(n+1)]/2 + + + +*Pour n = 1:* (1,k=1)Σk = 1 ; [n(n+1)]/2 = 1 . *So P(1) is true* + + + +For n ≥ 1. We assume that P(n) is true, OR : *(n, k=1)Σk = n(n+1)/2*. We now have to prove that P(n+1) is true, Or : *(n+1, k=1)Σk = (n+1)(n+2)/2* + + +(n+1, k=1)Σk = 1 + 2 + .... + n + (n+1) ; (n+1, k=1)Σk = (n, k=1)Σk + (n+1) ; = n(n+1)/2 + (n+1) ; = [n(n+1) + 2(n+1)]/2 ; = *[(n+2)(n+1)]/2* /WHICH IS WHAT WE NEEDED TO FIND/ + + +*Conclusion: ∀n ≥ 1 , (n,k=1)Σk = n(n+1)/2* + +* 4eme Cours : Chapitre 2 : Sets and Operations +** Definition of a set : +A set is a collection of objects that share the sane propriety + +** Belonging, inclusion, and equality : +a. Let E be a set. If x is an element of E, we say that x belongs to E we write *x ∈ E*, and if it doesn't, we write *x ∉ E* +b. A set E is included in a set F if all elements of E are elements of F and we write *E ⊂ F ⇔ (∀x , x ∈ E ⇒ x ∈ F)*. We say that E is a subset of F, or a part of F. The negation of this propriety is : *E ⊄ F ⇔ ∃x , x ∈ E and x ⊄ F* +c. E and F are equal if E is included in F and F is included in E, and we write *E = F ⇔ (E ⊂ F) et (F ⊂ E)* +d. The empty set (symbolized by ∅) is a set without elements, and is included in all sets (by convention) : *∅ ⊂ E* + +** Intersections and reunions : +*** Intersection: +E ∩ F = {x / x ∈ E AND x ∈ F} ; x ∈ E ∩ F ⇔ x ∈ F AND x ∈ F + + +x ∉ E ∩ F ⇔ x ∉ E OR x ∉ F + +*** Union: +E ∪ F = {x / x ∈ E OR x ∈ F} ; x ∈ E ∪ F ⇔ x ∈ F OR x ∈ F + + +x ∉ E ∪ F ⇔ x ∉ E AND x ∉ F +*** Difference between two sets: +E\F(Which is also written as : E - F) = {x / x ∈ E and x ∉ F} +*** Complimentary set: +If F ⊂ E. E - F is the complimentary of F in E. + + +FCE = {x /x ∈ E AND x ∉ F} *ONLY WHEN F IS A SUBSET OF E* +*** Symentrical difference +E Δ F = (E - F) ∪ (F - E) ; = (E ∪ F) - (E ∩ F) +** Proprieties : +Let E,F and G be 3 sets. We have : +*** Commutativity: +E ∩ F = F ∩ E +E ∪ F = F ∪ E +*** Associativity: +E ∩ (F ∩ G) = (E ∩ F) ∩ G +E ∪ (F ∪ G) = (E ∪ F) ∪ G +*** Distributivity: +E ∩ (F ∪ G) = (E ∩ F) ∪ (E ∩ G) +E ∪ (F ∩ G) = (E ∪ F) ∩ (E ∪ G) +*** Lois de Morgan: +If E ⊂ G and F ⊂ G ; + +(E ∩ F)CG = ECG ∪ FCG ; (E ∪ F)CG = ECG ∩ FCG +*** An other one: +E - (F ∩ G) = (E-F) ∪ (E-G) ; E - (F ∪ G) = (E-F) ∩ (E-G) +*** An other one: +E ∩ ∅ = ∅ ; E ∪ ∅ = E +*** And an other one: +E ∩ (F Δ G) = (E ∩ F) Δ (E ∩ G) +*** And the last one: +E Δ ∅ = E ; E Δ E = ∅ diff --git a/src/org/uni_notes/alsd1.org b/src/org/uni_notes/alsd1.org new file mode 100755 index 0000000..2619592 --- /dev/null +++ b/src/org/uni_notes/alsd1.org @@ -0,0 +1,33 @@ +#+title: ALSD1 +#+AUTHOR: Crystal +#+OPTIONS: ^:{} +#+OPTIONS: num:nil +#+EXPORT_FILE_NAME: ../../../uni_notes/alsd.html +#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../src/css/colors.css"/> +#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../src/css/style.css"/> +#+OPTIONS: html-style:nil +#+OPTIONS: toc:nil + +* Contenu de la Matiére +** Chapitre 1: Elements de Base +- Algorithmique, procésseur, action. +- Programme et languages de programmation. +- Analyse des problémes. + +** Chapitre 2: Présentation du formalisme Algorithmique +** Chapitre 3: Eléments de base du language C +** Chapitre 4: Modularité( Fonction et Procédure ) +** Chapitre 5: Les structures des données statiques + + +* Premier cours : Algorithmes /Oct 1/ : +** Définition d'un algorithm : +Un ensemble d'opérations ecrites dans le language naturel. + +*** Example d'un Algo : Résolution d'une équation du second ordre (ax²+bx+c=0) +1. Si a=0 ET b=0 alors *l'équation n'est pas du 2nd ordre*. +2. Si a=0 et b≠0 alors *x= -c/5* . +3. Si a≠0 alors *calculer Δ= b²-4ac* : + a. Si Δ=0 alors *x=-b/2a*. + b. Si Δ<0 alors *l'équation n'as pas de solution*. + c. Si Δ>0 alors *x=[-b±sqrt(Δ)]/2a* diff --git a/src/org/uni_notes/analyse1.org b/src/org/uni_notes/analyse1.org new file mode 100755 index 0000000..e511781 --- /dev/null +++ b/src/org/uni_notes/analyse1.org @@ -0,0 +1,477 @@ +#+title: Analyse 1 +#+AUTHOR: Crystal +#+OPTIONS: ^:{} +#+OPTIONS: num:nil +#+EXPORT_FILE_NAME: ../../../uni_notes/analyse.html +#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../src/css/colors.css"/> +#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../src/css/style.css"/> +#+OPTIONS: html-style:nil +#+OPTIONS: toc:nil + +* Contenu de la Matiére +** Chapitre 1 : Quelque propriétés de ℝ +- Structure algébrique de ℝ +- L'ordre dans ℝ +- Majorant, minorant, borne superieure, borne inférieure + +** Chapitre 2 : Les suites numériques réelles +- Définition : convergence, opérations sur les suites convergentes +- Theoréme de convergence, Theoréme de ___ suites, sans suites, extension au limites infinies +- Suites de cauchy, suites adjacentes et suites récurentes + +** Chapitre 3 : Limites et continuité des fonctions réelles d'une variable réelle +- Les limites : définition, opérations sur les limites, les formes inditerminées +- La continuité : définition, Theorémes fondamentaux +- La continuité informe les fonctions Lepchitziennes + +** Chapitre 4 : La dérivabilité et son interprétation géometrique +- Opérations sur les fonctions dérivales, Theoréme de Rolle, Theoréme des accroissements finis, régle de L'Hopital et formule de Taylor + +** Chapitre 5 : Les fonctions trigonométriques réciproques, fonctions hypérboliques réciproques +- Comparaison asymptotique +- Symbole de lamdau (lambda ?), et notions des fonctions équivalentes +- Développements limites polynominaux (D.L) et opérations sur les D.L +- Généralisations des D.L +- Application au calcul de limite et l'étude des branches infinies + + +* Premier cours : Quelque propriétés de ℝ /Sep 26/ : + +** La loi de composition interne dans E : +@ : E x E ---> E + (x,y) ---> x @ y + +@ est une lois de composition interne seulement si : + +*∀ x,y ε E* + + +*** *Example : Addition* +Est ce que l'addition (+) est L.C.I dans ℕ ? + +ℕ x ℕ ---> ℕ + +(x,y) ---> x + y ? /En gros : Pour que l'addition soit une L.C.I dans ℕ, il faut que: quand on additionne *n'importe quel* chiffre x et y de N, il faut que le résultat appertiens aussi a ℕ/ + +∀ x,y ∈ ℕ , x + y ∈ ℕ /En gros: Pour TOUTE valeur de x et y appartenant a ℕ, leur somme est toujours dans ℕ/ + +Donc : + est L.C.I dans ℕ + +*** *Example : soustraction* +Est ce que la soustraction (-) est L.C.I dans ℕ? + +ℕ x ℕ ---> ℕ + +(x,y) ---> x - y ? + + +∃ x , y ∈ ℕ , x - y ∉ ℕ /En gros: il existe au moins une valeur de x et y dans ℕ tel que leur différence n'est *PAS* dans ℕ . tel que : si x est 5, et y c'est 9. Leur différence est -4, qui appartiens pas a ℕ/ + +** La loi de composition externe dans E : + +@ est L.C.E dans E, K est un corps + +K x E ---> E + +(a,x) ---> a @ x + +∀ (a , x) ∈ K x E , a @ x ∈ E + +** Groupes : +/Soit E un ensemble, soit @ une L.C.I dans E/ + +(E, @) est un groupe Si : + +*** Il contiens un élement neutre + +∀ x ∈ E ; ∃ e ∈ E + +x @ e = e @ x = x + +On appelle *e* élement neutre + +/Ex: (ℕ,+) accepte un élement neutre, qui est 0, parceque x + 0 = 0 + x = x....cependent (ℕ,+) n'est pas un groupe. La raison est dans la prochaine condition/ + +*** Il contiens un élément symétrique + +∀ x ∈ E ; ∃ x' ∈ E ; x @ x' = x' @ x = e + +On appelle *x'* élèment symétrique + +/Dans l'example en haut, on remarque qu'il n'y ya pas de chiffre x' pour chaque chiffre x, qui est, l'hors de leur addition est egal a e (0), tout simplement car:/ + +/x + x' = e ; x + x' = 0 ; x = -x'/ + +*Or, Dans ℕ, on a pas de nombres négatifs* + +*** @ est cummutative : + +∀ (x , x') ∈ E x E ; x @ x' = x' @ x + +/L'addition est cummutative, la soustraction ne l'es pas. 5 + 3 ou 3 + 5 est pareil, mais 5 - 3 et 3 - 5 sont différents/ + +** Anneaux : +Soit E un ensemble, (E , @ , !) est un anneau si : +*** (E ; @) est un groupe cummutatif +*** ! est une loi associative : +∀ x , y , z ∈ E + +(x ! y) ! z = x ! (y ! z) +*** Distribution de ! par rapport à @ : +∀ x , y , z ∈ E + +(x @ y) ! z = ( x ! z ) @ ( y ! z ) +*** L'existance d'un élèment neutre de ! : + +∀ x ∈ E , ∃ e ∈ E , x ! e = e ! x = x + +*** ! est cummutative : + +∀ x , y ∈ E , x ! y = y ! x + +** Corps : +(E , @ , !) est un corps si les 5 conditions en haut sont vérifiées + cette condition : + +*** La symétrie : + +∀ x ∈ E ; ∃ x' ∈ E , x ! x' = x' ! x = e + +x' est l'élément symétrique de x par rapport à ! +(sauf élément neutre première lois ) + + +** Exercice : (ℝ, +, x) corps ou pas ? +*** Est-ce un Anneau ? +- (ℝ, +) est un groupe commutatif +- x est une loi associative : (a x b) x c = a x (b x c) +- On peut distribuer x par rapport a + : (a + b) x c = (a x c) + (b x c) +- Il existe un élément neutre de x which is 1 : a x 1 = 1 x a = a +- La multiplication est commutative : a x b = b x a + +Oui c'est un anneau +*** Est-ce un corps ? +- Oui : ∀ x ∈ ℝ\{e} ; x * x' = 1 + + +* 2nd cours :L'ordre dans ℝ, Majorant, minorant, borne superieure, borne inférieure /Oct 3/ : +** L'ordre dans ℝ +(ℝ, +, x) est un corps, Soit R une relation d'ordre dans ℝ si : + +1. R est antisymétrique : + + ∀ x, y ℝ ; (x R y et y R x) ⇒ (x = y) + +2. R est reflexive : + + ∀ x ∈ ℝ ; x R x + +3. R est transitive : + ∀ x, y, z ∈ ℝ , (x R y and y R z) ⇒ x R z + +*** Exemples : +**** Exemple numéro 1: +(ℝ , +, x) est un corps. Est ce la relation < est une relation d'ordre dans ℝ ? + + +Non, pourquoi ? parce que elle est pas réflexive : ∀ x ∈ ℝ, x < x **is obviously false** +**** Exemple numéro 2: +(ℝ , +, x) est un corps. Est ce la relation ≥ est une relation d'ordre dans ℝ ? + +1. (Antisymétrique) ∀ x, y ℝ ; (x ≥ y AND y ≥ x) ⇒ x = y is true +2. (Réflexive) ∀ x, y ℝ ; x ≥ x is true +3. (Transitive) ∀ x, y, z ℝ ; (x ≥ y AND y ≥ z) ⇒ x ≥ z is also true + +** Majorant, minorant, borne supérieure, borne inférieure +*** Majorant: +Soit E un sous-ensemble de ℝ (E ⊆ ℝ) + + +Soit a ∈ ℝ, a est un majorant de E Si :∀ x ∈ E , x ≤ a + +*** Minorant: +Soit E un sous-ensemble de ℝ (E ⊆ ℝ) + + +Soit b ∈ ℝ, b est un minorant de E Si :∀ x ∈ E , x ≥ b + + +*** Borne supérieure: +La borne supérieure est le plus petit des majorants /Sup(E) = Borne supérieure/ + +*** Borne inférieure: +La borne inférieure est le plus grand des minorant /Inf(E) = Borne inférieure/ + +*** Maximum : +E ⊆ ℝ, a est un maximum de E (Max(E)) Si : a ∈ E ; ∀x ∈ E, x ≤ a. +*** Minimum : +E ⊆ ℝ, b est un minimum de E (Min(E)) Si : b ∈ E ; ∀x ∈ E, x ≥ b. + +*** Remarques : +A et B deux ensembles bornés (Minoré et Majoré) : +1. A ∪ B est borné +2. A ∩ B est borné +3. Sup(A ∪ B)= Max(sup A, sup B) +4. Inf(A ∩ B)= Min(inf A, inf B) +5. Sup(A ∩ B)= Min(sup A, sup B) /Le plus petit des Supérieur de A et B/ +6. Inf(A ∩ B)= Max(inf A, inf B) /Le plus grand des inférieur de A et B/ + +* 3rd cours :Les suites numériques /Oct 5/ : +*** Définition : +Soit (Un)n ∈ ℕ une suite numérique , (Un)n est une application de ℕ dans ℝ: + + +ℕ ----> ℝ + + +n ----> U(n) = Un + +1. (Un) ou (Un)n ∈ ℝ : une suite +2. Un : terme général + +***** Exemple : +U : ℕ* ----> ℝ + + + n ----> 1/n + + + (Un) est une suite définit par Un = 1/n + +*** Définition N°2 : +On peut définir une suite â partir d'une relation de récurrence entre deux termes successifs et le premier terme. +***** Exemple : +U(n+1) = Un /2 + + +U(1)= 1 +** Opérations sur les suites : +*** La somme : +Soient (Un) et (Vn) deux suites, la somme de (Un) et (Vn) est une suite de terme général Un + Vn +*** Le produit : +Soient (Un)n et (Vn)n deux suites alors (Un) x (Vn) est une autre suite de terme général Un x Vn +*** Inverse d'une suite : +Soit Un une suite de terme général Un alors l'inverse de (Un) est une autre suite (Vn) = 1/(Un) de terme général de Vn = 1/Un +*** Produit d'une suite par un scalaire : +Soit (Un) une suite de T.G Un + + +∀ λ ∈ ℝ , λ(Un) n ∈ ℕ est une suite de T.G Vn= λUn + +** Suite bornée : +Une suite (Un) est bornée si (Un) majorée et minorée +** Suite majorée : +Soit (Un) une suite + + +U : (Un) est majorée par M ∈ ℝ ; ∀ n ∈ ℕ ; ∃ M ∈ ℝ , Un ≤ M + +** Suite minorée : +Soit (Un) une suite + + +U : (Un) est minorée par M ∈ ℝ ; ∀ n ∈ ℕ ; ∃ M ∈ ℝ , Un ≥ M + +** Suites monotones : +*** Les suites croissantes : +Soit (Un)n est une suite + + +(Un) est croissante si : ∀ n ∈ ℕ ; U(n+1) - Un ≥ 0 ⇔ Un+1 ≥ Un + +*** Les suites décroissantes : +Soit (Un)n est une suite + + +(Un) est décroissante si : ∀ n ∈ ℕ ; U(n+1) - Un ≤ 0 ⇔ Un+1 ≤ Un + + +* Série TD N°1 : /Oct 6/ +** Exo 1 : +*** Ensemble A : + +A = {-1/n , n ∈ ℕ *} + +**** Borne inférieure +∀ n ∈ ℕ* , -1/n ≥ -1 . -1 est la borne inférieure de l'ensemble A + +**** Minimum : +∀ n ∈ ℕ* , -1/n ≥ -1 . -1 est le Minimum de l'ensemble A + +**** Borne supérieure : +∀ n ∈ ℕ* , -1/n ≤ 0 . 0 est la borne supérieure de l'ensemble A +**** Maximum : +L'ensemble A n'as pas de maximum +*** Ensemble B : +B = [-1 , 3[ ∩ ℚ +**** Borne inférieure : +Inf(B) = Max(inf([-1 , 3[) , inf(ℚ)) + + +Puisse que ℚ n'as pas de Borne inférieure, donc par convention c'est *-∞*, + + +*Inf(B) = -1* + +**** Borne supérieure : +Sup(B) = Min(sup([-1 ,3[) , sup(ℚ)) + + +Puisse que ℚ n'as pas de Borne supérieure, donc par convention c'est *+∞*, + + +*Sup(B) = 3* + +**** Minimum : +*Min(B) = -1* + +**** Maximum : +L'ensemble B n'as pas de Maximum +*** Ensemble C : +C = {3n ,n ∈ ℕ} + +**** Borne inférieure : +Inf(C) = 0 + +**** Borne supérieure : +Sup(C) = +∞ +**** Minimum : +Min(C) = 0 +**** Maximum : +L'ensemble C n'as pas de Maximum + +*** Ensemble D : +D = {1 - 1/n , n ∈ ℕ*} +**** Borne inférieure : +Inf(D)= 0 +**** Borne supérieure : +Sup(D)= 1 +**** Minimum : +Min(D)= 0 +**** Maximum : +L'ensemble D n'as pas de Maximum + +*** Ensemble E : +E = { [2n + (-1)^n]/ n + 1 , n ∈ ℕ } + + +*Les valeurs que E peut prendre sont : "(2n + 1)/(n+1)" Si n est pair, et "(2n - 1)/(n+1)" si n est impair* + + +*On définit un ensemble F et G : F = { (2n + 1)/ (n+1) , n ∈ 2k}, G = { (2n - 1)/(n+1), n ∈ 2k+1}* + + +*Donc E = F ∪ G* + +**** Borne inférieure : +Inf(E) = Min(inf(F), inf(G)) + + +Inf(F) = 1 ; Inf(G) = -1 + + +*Inf(E)= -1* + +**** Borne supérieure : +Sup(E) = Max(sup(F), sup(G)) + + +sup(F) = +∞ ; sup(G) = +∞ + + +*Sup(E)= +∞* +**** Minimum : +Min(E)= -1 +**** Maximum : +E n'as pas de maximum +** Exo 2 : +*** Ensemble A : +A = {x ∈ ℝ , 0 < x <√3} + +**** Borné +*Oui*, Inf(A)= 0 ; Sup(A)=√3 +*** Ensemble B : +B = { x ∈ ℝ , 1/2 < sin x <√3/2} ; +**** Borné +*∀ x ∈ B, sin x > 1/2 ∴ Inf(B)= 1/2* + + +*∀ x ∈ B, sin x < √3/2 ∴ Sup(B)= √3/2* +*** Ensemble C : +C = {x ∈ ℝ , x³ > 3} +**** Minoré +*∀ x ∈ C, x³ > 3 ∴ Inf(C)= 3* +*** Ensemble D : +D = {x ∈ ℝ , e^x < 1/2} +**** Borné +*∀ x ∈ C, e^x > 0 ∴ Inf(C)= 0* + + +*∀ x ∈ C, e^x < 1/2 ∴ Sup(C)= 1/2* +*** Ensemble E : +E = {x ∈ ℝ , ∃ p ∈ ℕ* : x = √2/p} +**** Majoré +p = √2/x . Donc : *Sup(E)=1* +** Exo 3 : +U0 = 3/2 ; U(n+1) = (Un - 1)² + 1 +*** Question 1 : +Montrer que : ∀ n ∈ ℕ , 1 < Un < 2 . + + +*(Un - 1)² ≥ 0 /Parce que c'est un carré/* + + +*(Un - 1)² + 1 > 1* ; *U(n+1) ≥ 1* + + +**** Raisonnement par récurrence : +P(n) : ∀ n ∈ ℕ ; 1 < Un < 2 + + +P(0) est vraie : 1 < 3/2 < 2 + + +On suppose que P(n) est vraie et on vérifie P(n+1) pour une contradiction + + +1< Un < 2 ; 0 < Un - 1 < 1 ; 0 < (Un - 1)² < 1 ; 1 < (Un - 1)² + 1< 2 ; *1 < U(n+1) < 2* Donc elle est correcte + +*** Question 2 : +Montrer que (Un)n est strictement monotone : + + +*U(n+1) - Un = (Un - 1)² + 1 - Un* ; *U(n+1) - Un = Un² + 1 - 2Un + 1 - Un* ; *U(n+1) - Un = Un² - 3Un + 2* + + +On étudie *Un² - 3Un + 2* sur l'intervalle ]1, 2[ : Un² - 3Un + 2 = 0 est une équation du 2nd ordre, *Δ = 1* , elle accepte deux solutions : Un = 1 et Un = 2 + + +On déduit que *Un² - 3Un + 2* est négatif sur [1 , 2] et positif en dehors, donc *∀ 1 < Un < 2 , Un² - 3Un + 2 < 0* ; *∀ 1 < Un < 2 , U(n+1) - Un < 0* ; *∀ 1 < Un < 2 , U(n+1) < Un* Donc (Un)n est une suite strictement monotonne décroissante +* 4th cours (Suite) : /Oct 10/ +** Les suites convergentes +Soit (Un)n est une suite convergente si lim Un n--> +∞ = l +*** Remarque : +1. Un est une suite convergente alors Un est bornee +2. Un est une suite convergente lim Un n---> +∞ = l ⇔ lim |Un| n---> +∞ = |l| +3. Un est une suite majoree et croissante ⇒ Un converge +4. Un est une suite minoree et decroissante ⇒ Un converge +5. Soient (Un) et (Vn) deux suites convergentes, alors + a. Un + Vn est convergente + b. Un * Vn est convergente + c. ∀λ ∈ ℝ , (λUn) converge +6. Soit Un est une suite bornee et soit Vn une suite. lim Vn n->+∞ = 0 Alors lim Vn * Un n-> +∞ = 0 +** Theoreme d'encadrement +Soient Un Vn et Wn trois suites ∀n ∈ ℕ, Un ≤ Vn ≤ Wn . et lim Un n->∞ = lim Wn n-> +∞ = l ⇒ lim Vn n-> +∞ = l +** Suites arithmetiques +Un est une suite arithmetique si : U(n+1) = Un + r ; r etant la raison de la suite +*** Forme general +*Un = U0 + nr* ; *Un = Up + (n - p)r* +*** Somme des n premiers termes +Un est une suite arithmetique, Sn = [(U0 + Un)(n + 1)]/2 + + +Sn = (n, k = 0)ΣUk est une somme partielle et lim Sn n->+∞ = k≥0ΣUk est une serie +** Suites geometriques +*** Forme general +*Un = U0 x r^n* +*** Somme des n premiers termes +n ∈ ℕ\{1} Sn = U0 (1 - r^(n+1))/1-r diff --git a/src/org/uni_notes/architecture1.org b/src/org/uni_notes/architecture1.org new file mode 100755 index 0000000..3dde87d --- /dev/null +++ b/src/org/uni_notes/architecture1.org @@ -0,0 +1,370 @@ +#+title: Architecture 1 +#+AUTHOR: Crystal +#+OPTIONS: ^:{} +#+OPTIONS: num:nil +#+EXPORT_FILE_NAME: ../../../uni_notes/architecture.html +#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../src/css/colors.css"/> +#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../src/css/style.css"/> +#+OPTIONS: html-style:nil +#+OPTIONS: toc:nil + + +* Premier cours : Les systémes de numération /Sep 27/ : +Un système de numération est une méthode pour représenter des nombres à l'aide de symboles et de règles. Chaque système, comme le décimal (base 10) ou le binaire (base 2), utilise une base définie pour représenter des valeurs numériques. Il est caractérisé par 3 entitiés mathématiques importantes: + + 1. Une base (genre 10, ou 2) + 2. Un ensemble de chiffres + 3. Des régles de représentations des nombres + +*** *Examples :* +/B10 est un systéme de numération caractérisé par:/ +- Base = 10 +- Un ensemble de chiffres : (0,1,2,3,4,5,6,7,8,9) + +/B16 est un autre systéme de numération caractérisé par:/ +- Base = 16 +- Un ensemble de chiffres : (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F) + + *Puisse-qu'on peut pas utiliser des nombres a deux chiffres, on utilise des lettres aprés 9, en leur donnant des valeurs tel que :* + + A : 10 ; B : 11 ; C : 12 ; D : 13 ; E : 14 ; F : 15 + +** Comment passer d'un systéme a base 10 a un autre + +On symbolise un chiffre dans la base x par : (Nombre)x + +*** Pour les chiffres entiers : +*On fait une division successive, on prends le nombre 3257 comme exemple, on veut le faire passer d'une base décimale á une base 16:* + + +(3257)10 ----> (?)16 + + +On dévise 3257 par 16, et les restants de la division serra la valeur en base16: + +3257/16 = 203 + *9* / 16 + +203/16 = 12 + *B* / 16 /REMARQUE, 11 N'APPARTIENS PAS A L'ENSEMBLE DES CHIFFRES EN BASE16, CE QUI VEUT DIRE QU'ON LE REMPLACE PAR SON EQUIVALENT, DANS CE CAS LA: *B*/ + +12/16 = 0 + *C* / 16 /Pareil ici, 12 n'existe pas, donc c'est C. Autre note : La division s'arréte quand le résultat de la division est nul/ + +**** *Conclusion:* + +(3257)10 ----> (CB9)16 + +*** Pour les chiffres non entiers : + +*On fait la division successive pour la partie entiére, et une multiplication successive pour la partie rationelle:* + +(3257,32)10 ----> (?)16 + +On a déja la partie entiére donc on s'occupe de la partie aprés la virgule: + +0,32 x 16 = *5*,12 + +0,12 x 16 = *1*,92 + +0,92 x 16 = *E*,72 /On a pas de 15 donc c'est un E/ + +0,72 x 16 = *B*,52 + +0,52 x 16 = *8*,32 + +0,32 x 16 = *5*,12 + +... + +/On s'arréte quand on trouve un chiffre entier, et si on trouve pas, on s'arréte quand on remarque une répetition, dans ce cas la, la séquance 51EB8 vas se répéter indéfiniment, donc on se contente d'écrire la partie qui se répéte avec une barre en haut/ + + +(3257,32)10 ----> (CB9, _51EB8_)16 + + +* 2nd cours : Les systèmes de numération (Suite) /Oct 3/ : +** Comment passer d'une base N a la base 10 : + +Prenons comme exemple le nombre (11210,0011)3 , chaque chiffre dans ce nombre a un rang qui commence par 0 au premier chiffre (a gauche de la virgule) et qui augmente d'un plus qu'on avance a gauche, et diminue si on part a droite. Dans ce cas la : + + +(11210,0011)3 ; le 0 est de rang 0, le 1 est de rang 1, le 2 est de rang 2, le 1 est de rang 3, le 1 est de rang 4. Et si on part du coté de la virgule, 0 est de rang -1, 0 est de rang -2, le 1 est de rang -3, et le 1 est de rang -4. + + +Et pour passer a la base 10, il suffit d'appliquer cette formule : *Chiffre x Base^(rang) + 2emeChiffre x Base^(rang)... etc*, donc dans notre example: + + +/0 x 3° + 1 x 3¹ + 2 x 3² + 1 x 3³ + 1 x 3^4 + 0 x 3¯¹ + 0 x 3¯² + 1 x 3¯³ + 1 x 3^(-4) ≈ (129,05)10/ + + +** Comment passer d'une base N a une base N^(n) : + +Si il ya une relation entre une base et une autre, on peut directement transformer vers cette base. + +*** Exemple : +Pour passer de la base 2 a la base 8 (8 qui est 2³) on découpe les chiffres 3 par 3 + + +(1 101 011, 011)2 ; Pour le dernier 1 qui est seul ~tout comme moi~ il suffit d'ajouter des 0 à gauche (car on peut) pour compléter le découpage. + + +(001 101 011, 011)2; Next step c'est de dessiner le tableau de conversion de la base 2 a la base 8 ( un tableau a 3 bits ) + + +| N | | | | +|---+---+---+---| +| 0 | 0 | 0 | 0 | +| 1 | 0 | 0 | 1 | +| 2 | 0 | 1 | 0 | +| 3 | 0 | 1 | 1 | +| 4 | 1 | 0 | 0 | +| 5 | 1 | 0 | 1 | +| 6 | 1 | 1 | 0 | +| 7 | 1 | 1 | 1 | + + +Pour remplir on a qu'a diviser les chiffres en deux, et mettre des 0 dans la première partie et des 1 dans la 2éme, et en faire de même pour les autres colonnes . + + +Maintenant il suffit de trouver l'équivalent de la base2 en base8 : + + +001 c'est 1 ; 101 c'est 5 ; 011 c'est 3 ; donc *(1101011,011)2 ---> (153,3)8* + +** L'arithmétique binaire : +*** L'addition : +0 + 0 = 0 On retiens 0 + + +1 + 0 = 1 On retiens 0 + + +0 + 1 = 1 On retiens 0 + + +1 + 1 = 0 On retiens 1 + + +1 + 1 + 1 = 1 On retiens 1 + + +Donc 0110 + 1101 = 10011 + +*** La soustraction : +0 - 0 = 0 On emprunt = 0 + + +1 - 0 = 1 On emprunt = 0 + + +0 - 1 = 1 On emprunt = 1 + + +1 - 1 = 0 On emprunt = 0 + + +** TP N°1 : +*** Exo1: +| Base 10 | Base 2 | Base 3 | Base 8 | Base 16 | +|----------+------------------+---------------+--------+---------| +| 22,75 | 10110,11 | 211, _20_ | 26,6 | F6,C | +| 684,125 | 1010101100,001 | 221100, _01_ | 1254,1 | 2AC,2 | +| 3931,625 | 111101011011,101 | 1101121, _12_ | 7533,5 | F5B,A | +| 52,38 | 110100,011 | 1221,101 | 64,3 | 34,6147 | +| 10,67 | 1010,101 | 23,5 | 12,5 | A,AB85 | + +**** (10110,11)2 +0 x 2° + 1 x 2¹ + 1 x 2² + 0 x 2³ + 1 x 2^(4) + 1 x 2¯¹ + 1 x 2¯² = (22.75)10 + + +***** (22,75)10 ----> (3) + + + +22/3 = 7 R *1* ; 7/3 = 2 R *1* ; 2/3 = 0 R *2* + + +0,75 x 3 = *2*.25 ; 0,25 x 3 = *0*.75 ..... + + +(22,75)10 ----> (211, _20_) + +***** (10110,11)2 ----> (8) +8 = 2³ ; (010 110,110)2 ----> (?)8 + + +En utilisant le tableau 3bits : + + +010 : 2 ; 110 : 6 ; 110 : 6 + + +(10110,11)2 ----> (26,6)8 + +***** (22,75)10 ----> (16) +22/16 = 1 R *6* ; 1/16 : 0 R *F* + + +0,75 x 16 = *C* + + +(22,75)10 ----> (F6,C)16 + +**** (1254,1)8 +4 x 8° + 5 x 8¹ + 2 x 8² + 1 x 8³ + 1 x 8¯¹ = (684,125)10 + +***** (1254,1)8 ----> (?)2 +En utilisant le tableau 3bits : + + +001 010 101 100,001 /We will get rid of the leading zeros/ + + +(1010101100,001)2 + +***** (684,125)10 ----> (?)3 +684/3 = 228 R *0* ; 228/3 = 76 R *0* ; 76/3 = 25 R *1* ; 25/3 = 8 R *1* ; 8/3 = 2 R *2* ; 2/3 = 0 R *2* + + +0,125 x 3 = *0*,375 ; 0,375 x 3 = *1*,125 + + +(221100, _01_)3 + +***** (684,125)10 ----> (?)16 +684/16 = 42 R *C* ; 42/16 = 2 R *A* ; 2/16 0 R *2* + + +0,125 x 16 = *2* + + +(2AC,2)16 + +**** (F5B,A)16 +11 x 16° + 5 x 16 + 15 x 16² + 10 x 16¯¹ = (3931,625)10 +***** (3931,625)10 ----> (8) +3931/8 = 491 R *3* ; 491/8 = 61 R *3* ; 61/8 = 7 R *5* ; 7/8 = 0 R *7* + + +0,625 x 8 = *5* + + +(7533,5)8 +***** (7533,5)8 ----> (2) + +En utilisant le tableau 3bits + +(111 101 011 011,101)2 + +***** (3931,625)10 ----> (3) +3931/3 = 1310 R *1* ; 1310/3 = 436 R *2* ; 436/3 = 145 R *1* ; 145/3 = 48 R *1* ; 48/3 = 16 R *0* ; 16/3 = 5 R *1* ; 5/3 = 1 R *2* ; 1/3 = 0 R *1* + + +0.625 x 3 = *1*,875 ; 0,875 x 3 = *2*,625 + + +(1101121, _12_)3 + +**** (52,38)10 +52/2 = 26 R *0* ; 26/2 = 13 R *0* ; 13/2 = 6 R *1* ; 6/2 = 3 R *0* ; 3/2 = 1 R *1* ; 1/2 = 0 R *1* + + +0,38 x 2 = *0*,76 ; 0,76 x 2 = *1*,52 ; 0,52 x 2 = *1*,04 ; 0,04 x 2 = *0*,08 .... + + +(110100,0110)2 + +***** (52,38)10 ----> (3) +52/3 = 17 R *1* ; 17/3 = 5 R *2* ; 5/3 = 1 R *2* ; 1/3 = 0 R *1* + + +0,38 x 3 = *1*.14 ; 0,14 x 3 = *0*.42 ; 0,42 x 3 = *1*.26 ; 0.26 x 3 = *0*.78 ... + + +(1221,101)3 + +***** (110100,011)2 ----> (8) +En utilisant le tableau 3bits: + + +(110 100,011)2 ----> (64,3)8 + +***** (52,38)10 ----> (16) +52/16 = 3 R *4* ; 3/16 = 0 R *3* + + +0,38 x 16 = *6*,08 ; 0,08 x 16 = *1*,28 ; 0,28 x 16 = *4*,48 ; 0,48 x 16 = *7*,68 .... + + +(34,6147)16 + +**** (23,5)3 +3 x 3° + 2 x 3 + 5 x 3¯¹ = (10.67)10 + +***** (10,67)10 ----> (2) +10/2 = 5 R *0* ; 5/2 = 2 R *1* ; 2/2 = 1 R *0* ; 1/2 = 0 R *1* + + +0,67 x 2 = *1*,34 ; 0,34 x 2 = *0*,68 ; 0,68 x 2 = *1*,36 ; 0,36 x 2 = *0*,72 ... + + +(1010,101)2 + +***** (001 010,101)2 ----> (8) +*Ô Magic 3bits table, save me soul, me children and me maiden:* + + +(12,5)8 +***** (10,67)10 ----> (16) +10/16 = 0 R *A* + + +0,67 x 16 = *A*,72 ; 0,72 x 16 = *B*,52 ; 0,52 x 16 = *8*,32 ; 0,32 x 16 = *5*,12 ... + + +(A,AB85)16 + +*** Exo2: +**** (34)? = (22)10 +(34)a = (22)10 ; 4 x a° + 3 x a = 22 ; 4 + 3a = 22 ; 3a = 18 + + +*a = 6* +**** (75)? = (117)10 +(75)b = (117)10 ; 5 x b° + 7 x b¹ = 117 ; 5 + 7b = 117 ; 7b = 112 + + +*b = 16* +*** Exo3: +**** (101011)2 + (111011)2 + 101011 + 111011 = 1100110 +**** (1011,1101)2 + (11,1)2 + 1011,1101 + 11,1000 = 1111,0101 +**** (1010,0101)2 - (110,1001)2 +1010,0101 - 110,1001 = 11,1100 +** L'arithmétique binaire (Suite): /Oct 4/ +*** La multiplication : +0 x 0 = 0 + + +0 x 1 = 0 + + +1 x 0 = 0 + + +1 x 1 = 1 + +*** La division : +On divise de la manière la plus normale du monde !!! +* 4th cours : Le codage /Oct 10/ +** Le codage des entiers positifs +Le codage sur n bits permet de representer tout les entiers naturels compris entre [0, 2^n - 1]. On peut coder sur 8bits les entiers entre [0;2^8 - 1(255)] +** Le codage des nombres relatifs +*** Remarque +Quelque soit le codage utilise, par convention le dernier bit est reserve pour le signe. ou 1 est negatif et 0 est positif. +*** Le codage en signe + valeur absolue (SVA): +Avec n bits le n eme est reserve au signe : [-(2^n-1)-1 , 2^n-1 -1]. Sur 8bits [-127, 127] + +*** Codage en compliment a 1 (CR): +On obtiens le compliment a 1 d'un nombre binaire en inversant chaqu'un de ses bits (1 -> 0 et 0-> 1) les nombres positifs sont la meme que SVA (il reste tel qu'il est) +*** Codage en compliment a 2 (CV): +C'est literallement CR + 1 pour les negatifs et SVA pour les nombres positifs diff --git a/src/txt/pubkey.asc b/src/txt/pubkey.asc new file mode 100755 index 0000000..ffaca42 --- /dev/null +++ b/src/txt/pubkey.asc @@ -0,0 +1,13 @@ +-----BEGIN PGP PUBLIC KEY BLOCK----- + +mDMEZLv2mhYJKwYBBAHaRw8BAQdAOiOQf7duuBZoGsrXWt68o+uNrKIGUDh5uepm +hdm6hDK0H0NyeXN0YWwgPGNyeXN0YWxAZGFud2luMTIxMC5kZT6IkwQTFgoAOxYh +BHwwpMu5L2H/puVOqXv3mQliAjaxBQJku/aaAhsDBQsJCAcCAiICBhUKCQgLAgQW +AgMBAh4HAheAAAoJEHv3mQliAjax2h4A/RnkHKtk1xWL/ryoKw9H6BmKRXAkdVeR +I18jEq67D8vFAP9JhrhnpiQln1S4JO3ZE/5e0Q9pY0EuO2LN/WECIUaDC7g4BGS7 +9poSCisGAQQBl1UBBQEBB0CCfjAXR0bVG2JYKACTjB1xRnUwGBLBSVc1+csWAOMn +YwMBCAeIeAQYFgoAIBYhBHwwpMu5L2H/puVOqXv3mQliAjaxBQJku/aaAhsMAAoJ +EHv3mQliAjaxNoABALuYhTgcRkxwZRYohmTfppcs+TwPmzQXkNoIk8qB6GxeAP0f +7yu7o+9Cap440VMX38gX3tZxZmx0SznrQgiZLvnBAg== +=mUPw +-----END PGP PUBLIC KEY BLOCK----- diff --git a/super_secret.html b/super_secret.html new file mode 100755 index 0000000..1bc8695 --- /dev/null +++ b/super_secret.html @@ -0,0 +1,38 @@ +<?xml version="1.0" encoding="utf-8"?> +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" +"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> +<head> +<!-- 2023-09-10 Sun 19:11 --> +<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> +<meta name="viewport" content="width=device-width, initial-scale=1" /> +<title>SUPER SECRET WEBSITE</title> +<meta name="author" content="Crystal" /> +<meta name="generator" content="Org Mode" /> +<link rel="stylesheet" type="text/css" href="src/css/colors.css"/> +<link rel="stylesheet" type="text/css" href="src/css/style.css"/> +</head> +<body> +<div id="content" class="content"> +<h1 class="title">SUPER SECRET WEBSITE</h1> +<div class="smurf"> + + +<div id="org6b38aea" class="figure"> +<p><img src="./src/gifs/smurf.jpg" alt="smurf.jpg" /> +</p> +</div> +</div> + + +<audio controls autoplay loop> + <source src="./src/media/welive.mp3" type="audio/mpeg"> + Your browser does not support the audio element. +</audio> +</div> +<div id="postamble" class="status"> +<p class="author">Author: Crystal</p> +<p class="date">Created: 2023-09-10 Sun 19:11</p> +</div> +</body> +</html> \ No newline at end of file diff --git a/uni_notes/algebra.html b/uni_notes/algebra.html new file mode 100755 index 0000000..27c8d76 --- /dev/null +++ b/uni_notes/algebra.html @@ -0,0 +1,1295 @@ +<?xml version="1.0" encoding="utf-8"?> +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" +"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> +<head> +<!-- 2023-10-11 Wed 19:04 --> +<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> +<meta name="viewport" content="width=device-width, initial-scale=1" /> +<title>Algebra 1</title> +<meta name="author" content="Crystal" /> +<meta name="generator" content="Org Mode" /> +<link rel="stylesheet" type="text/css" href="../src/css/colors.css"/> +<link rel="stylesheet" type="text/css" href="../src/css/style.css"/> +<script> + window.MathJax = { + tex: { + ams: { + multlineWidth: '85%' + }, + tags: 'ams', + tagSide: 'right', + tagIndent: '.8em' + }, + chtml: { + scale: 1.0, + displayAlign: 'center', + displayIndent: '0em' + }, + svg: { + scale: 1.0, + displayAlign: 'center', + displayIndent: '0em' + }, + output: { + font: 'mathjax-modern', + displayOverflow: 'overflow' + } + }; +</script> + +<script + id="MathJax-script" + async + src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"> +</script> +</head> +<body> +<div id="content" class="content"> +<h1 class="title">Algebra 1</h1> +<div id="outline-container-orgcc5f4d8" class="outline-2"> +<h2 id="orgcc5f4d8">Contenu de la Matiére</h2> +<div class="outline-text-2" id="text-orgcc5f4d8"> +</div> +<div id="outline-container-orgf4040f2" class="outline-3"> +<h3 id="orgf4040f2">Rappels et compléments (11H)</h3> +<div class="outline-text-3" id="text-orgf4040f2"> +<ul class="org-ul"> +<li>Logique mathématique et méthodes du raisonnement mathématique</li> +<li>Ensembles et Relations</li> +<li>Applications</li> +</ul> +</div> +</div> +<div id="outline-container-orge74cdc9" class="outline-3"> +<h3 id="orge74cdc9">Structures Algébriques (11H)</h3> +<div class="outline-text-3" id="text-orge74cdc9"> +<ul class="org-ul"> +<li>Groupes et morphisme de groupes</li> +<li>Anneaux et morphisme d’anneaux</li> +<li>Les corps</li> +</ul> +</div> +</div> +<div id="outline-container-org0b54650" class="outline-3"> +<h3 id="org0b54650">Polynômes et fractions rationnelles</h3> +<div class="outline-text-3" id="text-org0b54650"> +<ul class="org-ul"> +<li>Notion du polynôme à une indéterminée á coefficients dans un anneau</li> +<li>Opérations Algébriques sur les polynômes</li> +<li>Arithmétique dans l’anneau des polynômes</li> +<li>Polynôme dérivé et formule de Taylor</li> +<li>Notion de racine d’un polynôme</li> +<li>Notion de Fraction rationelle á une indéterminée</li> +<li>Décomposition des fractions rationelles en éléments simples</li> +</ul> +</div> +</div> +</div> +<div id="outline-container-org9dbc8bb" class="outline-2"> +<h2 id="org9dbc8bb">Premier cours : Logique mathématique et méthodes du raisonnement mathématique <i>Sep 25</i> :</h2> +<div class="outline-text-2" id="text-org9dbc8bb"> +<p> +Let <b>P</b> <b>Q</b> and <b>R</b> be propositions which can either be <b>True</b> or <b>False</b>. And let’s also give the value <b>1</b> to each <b>True</b> proposition and <b>0</b> to each false one. +</p> + +<p> +<i>Ex:</i> +</p> +<ul class="org-ul"> +<li><b>5 ≥ 2</b> is a proposition, a correct one !!!</li> +<li><b>The webmaster is a girl</b> is also a proposition, which is also correct.</li> +<li><b>x is always bigger than 5</b> is <b>not</b> a proposition, because we CAN’T determine if it’s correct or not as <b>x</b> changes.</li> +</ul> +<p> +…etc +</p> + +<p> +In order to avoid repetition, and rewriting the proposition over and over, we just assign a capital letter to them such as <b>P Q</b> or <b>R</b>. +</p> + +<p> +So now we could write : +<b>Let the proposition P be 5 ≥ 2, we notice that P is always True, therefor its validity is 1</b> +</p> + +<p> +We also have the opposite of <b>P</b>, which is <b>not(P)</b> but for simplicity we use <b>P̅</b> (A P with a bar on top, in case it doesn’t load for you), now let’s go back to the previous example: +</p> + +<p> +<b>Since we know that the proposition P is true, we can conclude that P̅ is false. As P and P̅ can NOT be true at the same time. It’s like saying 5 is greater and also lesser than 2…doesn’t make sense, does it ?</b> +</p> + +<p> +Now let’s say we have two propositions, and we want to test the validity of their disjunction….. Okay what is this “disjunction” ? <b>Great Question Billy !!!</b> A disjunction is true if either propositions are true +</p> + +<p> +Ex: +<b>Let proposition P be “The webmaster is asleep”, and Q be “The reader loves pufferfishes”. The disjunction of these two propositions can have 4 different values showed in this Table of truth (such a badass name):</b> +</p> + +<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> + + +<colgroup> +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> +</colgroup> +<thead> +<tr> +<th scope="col" class="org-right">P</th> +<th scope="col" class="org-right">Q</th> +<th scope="col" class="org-right">Disjunction</th> +</tr> +</thead> +<tbody> +<tr> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +</tr> + +<tr> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +</tr> + +<tr> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +</tr> + +<tr> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +</tr> +</tbody> +</table> + +<p> +<i>What the hell is this ?</i> +The first colomn is equivalent to saying : “The webmaster is asleep AND The reader loves pufferfishes” +The second one means : “The webmaster is asleep AND The reader DOESN’T love pufferfishes (if you are in this case, then <b>I HATE YOU</b>)” +The third one… <i>zzzzzzz</i> +</p> + +<p> +You got the idea !!! +And since we are talking about a disjunction here, <b>one of the propositions</b> need to be true in order for this disjunction to be true. +</p> + +<p> +You may be wondering…. Crystal, can’t we write a disjunction in magical math symbols ? And to this I respond with a big <b>YES</b>. A disjunction is symbolized by a <b>∨</b> . So the disjunction between proposition <b>P & Q</b> can be written this way : <b>P ∨ Q</b> +</p> + +<p> +What if, we want to test whether or not two propositions are true AT THE SAME TIME ? Long story short, we can, it’s called a conjunction, same concept, as before, only this time the symbol is <b>P ∧ Q</b>, and is only true if <b>P</b> and <b>Q</b> are true. So we get a Table like this : +</p> + +<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> + + +<colgroup> +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> +</colgroup> +<thead> +<tr> +<th scope="col" class="org-right">P</th> +<th scope="col" class="org-right">Q</th> +<th scope="col" class="org-right">P ∨ Q</th> +<th scope="col" class="org-right">P ∧ Q</th> +</tr> +</thead> +<tbody> +<tr> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +</tr> + +<tr> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +</tr> + +<tr> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +</tr> + +<tr> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +</tr> +</tbody> +</table> + +<p> +<b>Always remember: 1 means true and 0 means false</b> +</p> + +<p> +There are two more basics to cover here before going to some properties, the first one is implication symbolized by the double arrow <b>⇒</b> +</p> + +<p> +Implication is kinda hard for my little brain to explain, so I will just say what it means: +</p> + +<p> +<b>If P implies Q, this means that either Q, or the opposite of P are correct</b> +</p> + +<p> +or in math terms +</p> + +<p> +<b>P ⇒ Q translates to P̅ ∨ Q</b> +Let’s illustrate : +</p> + +<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> + + +<colgroup> +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> +</colgroup> +<thead> +<tr> +<th scope="col" class="org-right">P</th> +<th scope="col" class="org-right">Q</th> +<th scope="col" class="org-right">P̅</th> +<th scope="col" class="org-right">Q̅</th> +<th scope="col" class="org-right">P ∨ Q</th> +<th scope="col" class="org-right">P ∧ Q</th> +<th scope="col" class="org-right">P ⇒ Q (P̅ ∨ Q)</th> +</tr> +</thead> +<tbody> +<tr> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +</tr> + +<tr> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +</tr> + +<tr> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +</tr> + +<tr> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +</tr> +</tbody> +</table> + +<p> +<b>If you look clearly, there is only one case where an implication is false. therefor you just need to find it, and blindly say that the others are correct. A rule of thumb is that: “A correct never implies a false”, or “If a 1 tries to imply a 0, the implication is a 0”</b> +</p> + +<p> +Aight, a last one and we are done!!! Equivalence, which is fairly easy, symbolized by a <b>⇔</b> symbol. +</p> + +<p> +A proposition is equivalent to another only when both of them have <b>the same value of truth</b> AKA: both true or both false. a little table will help demonstrate what i mean. +</p> + +<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> + + +<colgroup> +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> +</colgroup> +<thead> +<tr> +<th scope="col" class="org-right">P</th> +<th scope="col" class="org-right">Q</th> +<th scope="col" class="org-right">P̅</th> +<th scope="col" class="org-right">Q̅</th> +<th scope="col" class="org-right">P ∨ Q</th> +<th scope="col" class="org-right">P ∧ Q</th> +<th scope="col" class="org-right">P ⇒ Q (P̅ ∨ Q)</th> +<th scope="col" class="org-right">P ⇔ Q</th> +</tr> +</thead> +<tbody> +<tr> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +</tr> + +<tr> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +</tr> + +<tr> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +</tr> + +<tr> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +</tr> +</tbody> +</table> + +<p> +<i>Note: P implying Q is equivalent to P̅ implying Q̅, or: (P ⇒ Q) ⇔ (P̅ ⇒ Q̅)</i> +</p> +</div> +<div id="outline-container-org29099d5" class="outline-3"> +<h3 id="org29099d5">Properties:</h3> +<div class="outline-text-3" id="text-org29099d5"> +</div> +<div id="outline-container-orgfeb62c8" class="outline-4"> +<h4 id="orgfeb62c8"><b>Absorption</b>:</h4> +<div class="outline-text-4" id="text-orgfeb62c8"> +<p> +(P ∨ P) ⇔ P +</p> + +<p> +(P ∧ P) ⇔ P +</p> +</div> +</div> +<div id="outline-container-org2fa2b1f" class="outline-4"> +<h4 id="org2fa2b1f"><b>Commutativity</b>:</h4> +<div class="outline-text-4" id="text-org2fa2b1f"> +<p> +(P ∧ Q) ⇔ (Q ∧ P) +</p> + +<p> +(P ∨ Q) ⇔ (Q ∨ P) +</p> +</div> +</div> +<div id="outline-container-orge30be26" class="outline-4"> +<h4 id="orge30be26"><b>Associativity</b>:</h4> +<div class="outline-text-4" id="text-orge30be26"> +<p> +P ∧ (Q ∧ R) ⇔ (P ∧ Q) ∧ R +</p> + +<p> +P ∨ (Q ∨ R) ⇔ (P ∨ Q) ∨ R +</p> +</div> +</div> +<div id="outline-container-orgd1de8d5" class="outline-4"> +<h4 id="orgd1de8d5"><b>Distributivity</b>:</h4> +<div class="outline-text-4" id="text-orgd1de8d5"> +<p> +P ∧ (Q ∨ R) ⇔ (P ∧ Q) ∨ (P ∧ R) +</p> + +<p> +P ∨ (Q ∧ R) ⇔ (P ∨ Q) ∧ (P ∨ R) +</p> +</div> +</div> +<div id="outline-container-org8c5c9f6" class="outline-4"> +<h4 id="org8c5c9f6"><b>Neutral element</b>:</h4> +<div class="outline-text-4" id="text-org8c5c9f6"> +<p> +<i>We define proposition <b>T</b> to be always <b>true</b> and <b>F</b> to be always <b>false</b></i> +</p> + +<p> +P ∧ T ⇔ P +</p> + +<p> +P ∨ F ⇔ P +</p> +</div> +</div> +<div id="outline-container-orgf102dde" class="outline-4"> +<h4 id="orgf102dde"><b>Negation of a conjunction & a disjunction</b>:</h4> +<div class="outline-text-4" id="text-orgf102dde"> +<p> +Now we won’t use bars here because my lazy ass doesn’t know how, so instead I will use not()!!! +</p> + +<p> +not(<b>P ∧ Q</b>) ⇔ P̅ ∨ Q̅ +</p> + +<p> +not(<b>P ∨ Q</b>) ⇔ P̅ ∧ Q̅ +</p> + +<p> +<b>A rule I really like to use here is: Break and Invert. Basically you break the bar into the three characters of the propositions, so you get not(P) not(∧ or ∨) <i>NOT AN ACTUAL MATH WRITING. DONT USE IT ANYWHERE ELSE OTHER THAN YOUR BRAIN</i> and not(Q)</b> +</p> +</div> +</div> +<div id="outline-container-org9534ece" class="outline-4"> +<h4 id="org9534ece"><b>Transitivity</b>:</h4> +<div class="outline-text-4" id="text-org9534ece"> +<p> +[(P ⇒ Q) (Q ⇒ R)] ⇔ P ⇒ R +</p> +</div> +</div> +<div id="outline-container-org90fa987" class="outline-4"> +<h4 id="org90fa987"><b>Contraposition</b>:</h4> +<div class="outline-text-4" id="text-org90fa987"> +<p> +(P ⇒ Q) ⇔ (Q̅ ⇒ P̅) +</p> +</div> +</div> +<div id="outline-container-orga2d0ece" class="outline-4"> +<h4 id="orga2d0ece">God only knows what this property is called:</h4> +<div class="outline-text-4" id="text-orga2d0ece"> +<p> +<i>If</i> +</p> + +<p> +(P ⇒ Q) is true +</p> + +<p> +and +</p> + +<p> +(Q̅ ⇒ Q) is true +</p> + +<p> +then +</p> + +<p> +Q is always true +</p> +</div> +</div> +</div> +<div id="outline-container-org35a43b7" class="outline-3"> +<h3 id="org35a43b7">Some exercices I found online :</h3> +<div class="outline-text-3" id="text-org35a43b7"> +</div> +<div id="outline-container-orgf619324" class="outline-4"> +<h4 id="orgf619324">USTHB 2022/2023 Section B :</h4> +<div class="outline-text-4" id="text-orgf619324"> +</div> +<ul class="org-ul"> +<li><a id="org1c47389"></a>Exercice 1: Démontrer les équivalences suivantes:<br /> +<div class="outline-text-5" id="text-org1c47389"> +<ol class="org-ol"> +<li><p> +(P ⇒ Q) ⇔ (Q̅ ⇒ P̅) +</p> + +<p> +Basically we are asked to prove contraposition, so here we have ( P ⇒ Q ) which is equivalent to P̅ ∨ Q <b>By definition : (P ⇒ Q) ⇔ (P̅ ∨ Q)</b> +</p></li> +</ol> + + +<p> +So we end up with : <b>(P̅ ∨ Q) ⇔ (Q̅ ⇒ P̅)</b>, now we just do the same with the second part of the contraposition. <b>(Q̅ ⇒ P̅) ⇔ (Q ∨ P̅)</b> therefor : +</p> + + +<p> +<b>(Q ∨ P̅) ⇔ (P̅ ∨ Q)</b>, which is true because of commutativity +</p> + +<ol class="org-ol"> +<li>not(P ⇒ Q) ⇔ P ∧ Q̅</li> +</ol> + + +<p> +Okaaaay so, let’s first get rid of the implication, because I don’t like it : <b>not(P̅ ∨ Q)</b> +</p> + + +<p> +Now that we got rid of it, we can negate the whole disjunction <b>not(P̅ ∨ Q) ⇔ (P ∧ Q̅)</b>. Which is the equivalence we needed to prove +</p> + +<ol class="org-ol"> +<li><p> +P ⇒ (Q ∧ R) ⇔ (P ⇒ Q) ∧ (P ⇒ R) +</p> + +<p> +One might be tempted to replace P with P̅ to get rid of the implication…sadly this isnt it. All we have to do here is resort to <b>Distributivity</b>, because yeah, we can distribute an implication across a {con/dis}junction +</p></li> + +<li><p> +P ∧ (Q ∨ R) ⇔ (P ∧ Q) ∨ (P ∧ R) +</p> + +<p> +Literally the same as above 🩷 +</p></li> +</ol> +</div> +</li> +<li><a id="org688fdcc"></a>Exercice 2: Dire si les propositions suivantes sont vraies ou fausses, et les nier:<br /> +<div class="outline-text-5" id="text-org688fdcc"> +<ol class="org-ol"> +<li><p> +∀x ∈ ℝ ,∃y ∈ ℝ*+, tels que e^x = y +</p> + +<p> +For each x from the set of Real numbers, there exists a number y from the set of non-zero positive Real numbers that satisfies the equation : e^x = y +</p></li> +</ol> + + +<p> +“The function f(x)=e^x is always positive and non-null”, the very definition of an exponential function !!!! +</p> + + +<p> +<b>So the proposition is true</b> +</p> + + +<ol class="org-ol"> +<li>∃x ∈ ℝ, tels que x^2 < x < x^3</li> +</ol> + + +<p> +We just need to find a value that satisifies this condition…thankfully its easy…. +</p> + +<p> +x² < x < x³ , we divide the three terms by x so we get : +</p> + + +<p> +x < 1 < x² , or : +</p> + + +<p> +<b>x < 1</b> ; <b>1 < x²</b> ⇔ <b>x < 1</b> ; <b>1 < x</b> <i>We square root both sides</i> +</p> + + +<p> +We end up with a contradiction, therefor its wrong +</p> + + +<ol class="org-ol"> +<li>∀x ∈ ℝ, ∃y ∈ ℝ tels que y = 3x - 8</li> +</ol> + + +<p> +I dont really understand this one, so let me translate it “For any value of x from the set of Real numbers, 3x - 8 is a Real number”…. i mean….yeah, we are substracting a Real number from an other real number… +</p> + +<p> +<b>Since substraction is an Internal composition law in ℝ, therefor all results of a substraction between two Real numbers is…Real</b> +</p> + +<ol class="org-ol"> +<li><p> +∃x ∈ ℕ, ∀y ∈ ℕ, x > y ⇒ x + y < 8 +</p> + +<p> +“There exists a number x from the set of Natural numbers such as for all values of y from the set of Natural numbers, x > y implies x + y < 8” +</p></li> +</ol> + + +<p> +Let’s get rid of the implication : +</p> + +<p> +∃x ∈ ℕ, ∀y ∈ ℕ, (y > x) ∨ (x + y < 8) <i>There exists a number x from the set of Natural numbers such as for all values of y from the set of Natural numbers y > x OR x + y < 8</i> +</p> + +<p> +This proposition is true, because there exists a value of x that satisfies this condition, it’s <b>all numbers under 8</b> let’s take 3 as an example: +</p> + + +<p> +<b>x = 3 , if y > 3 then the first condition is true ; if y < 3 then the second one is true</b> +</p> + + +<p> +Meaning that the two propositions CAN NOT BE WRONG TOGETHER, either one is wrong, or the other +</p> + + +<p> +y > x +</p> + + +<p> +<b>y - x > 0</b> +</p> + + +<p> +y + x < 8 +</p> + + +<p> +<b>y < 8 - x</b> <i>This one is always true for all values of x below 8, since we are working in the set ℕ</i> +</p> + + +<ol class="org-ol"> +<li><p> +∀x ∈ ℝ, x² ≥ 1 ⇔ x ≥ 1 +</p> + +<p> +….This is getting stupid. of course it’s true it’s part of the definition of the power of 2 +</p></li> +</ol> +</div> +</li> +</ul> +</div> +</div> +</div> +<div id="outline-container-orgac834f2" class="outline-2"> +<h2 id="orgac834f2">2éme cours <i>Oct 2</i></h2> +<div class="outline-text-2" id="text-orgac834f2"> +</div> +<div id="outline-container-org42a8fad" class="outline-3"> +<h3 id="org42a8fad">Quantifiers</h3> +<div class="outline-text-3" id="text-org42a8fad"> +<p> +A propriety P can depend on a parameter x +</p> + + +<p> +∀ is the universal quantifier which stands for “For any value of…” +</p> + + +<p> +∃ is the existential quantifier which stands for “There exists at least one…” +</p> +</div> +<ul class="org-ul"> +<li><a id="org48c4773"></a>Example<br /> +<div class="outline-text-6" id="text-org48c4773"> +<p> +P(x) : x+1≥0 +</p> + +<p> +P(X) is True or False depending on the values of x +</p> +</div> +</li> +</ul> +<div id="outline-container-org923b5fe" class="outline-4"> +<h4 id="org923b5fe">Proprieties</h4> +<div class="outline-text-4" id="text-org923b5fe"> +</div> +<ul class="org-ul"> +<li><a id="orgaf93dd2"></a>Propriety Number 1:<br /> +<div class="outline-text-5" id="text-orgaf93dd2"> +<p> +The negation of the universal quantifier is the existential quantifier, and vice-versa : +</p> + +<ul class="org-ul"> +<li>not(∀x ∈ E , P(x)) ⇔ ∃ x ∈ E, not(P(x))</li> +<li>not(∃x ∈ E , P(x)) ⇔ ∀ x ∈ E, not(P(x))</li> +</ul> +</div> +<ul class="org-ul"> +<li><a id="orgf4f038f"></a>Example:<br /> +<div class="outline-text-6" id="text-orgf4f038f"> +<p> +∀ x ≥ 1 x² > 5 ⇔ ∃ x ≥ 1 x² < 5 +</p> +</div> +</li> +</ul> +</li> +<li><a id="org2c00dae"></a>Propriety Number 2:<br /> +<div class="outline-text-5" id="text-org2c00dae"> +<p> +<b>∀x ∈ E, [P(x) ∧ Q(x)] ⇔ [∀ x ∈ E, P(x)] ∧ [∀ x ∈ E, Q(x)]</b> +</p> + + +<p> +The propriety “For any value of x from a set E , P(x) and Q(x)” is equivalent to “For any value of x from a set E, P(x) AND for any value of x from a set E, Q(x)” +</p> +</div> +<ul class="org-ul"> +<li><a id="orga95ad05"></a>Example :<br /> +<div class="outline-text-6" id="text-orga95ad05"> +<p> +P(x) : sqrt(x) > 0 ; Q(x) : x ≥ 1 +</p> + + +<p> +∀x ∈ ℝ*+, [sqrt(x) > 0 , x ≥ 1] ⇔ [∀x ∈ R*+, sqrt(x) > 0] ∧ [∀x ∈ R*+, x ≥ 1] +</p> + + +<p> +<b>Which is true</b> +</p> +</div> +</li> +</ul> +</li> +<li><a id="orgfe5ddf2"></a>Propriety Number 3:<br /> +<div class="outline-text-5" id="text-orgfe5ddf2"> +<p> +<b>∃ x ∈ E, [P(x) ∧ Q(x)] <i>⇒</i> [∃ x ∈ E, P(x)] ∧ [∃ x ∈ E, Q(x)]</b> +</p> + + +<p> +<i>Here its an implication and not an equivalence</i> +</p> +</div> +<ul class="org-ul"> +<li><a id="orgca7bea7"></a>Example of why it’s NOT an equivalence :<br /> +<div class="outline-text-6" id="text-orgca7bea7"> +<p> +P(x) : x > 5 ; Q(x) : x < 5 +</p> + + +<p> +Of course there is no value of x such as its inferior and superior to 5 at the same time, so obviously the proposition is false. However, the two propositions separated are correct on their own, because there is a value of x such as its superior to 5, and there is also a value of x such as its inferior to 5. This is why it’s an implication and NOT AN EQUIVALENCE!!! +</p> +</div> +</li> +</ul> +</li> +<li><a id="orgf4ecdc0"></a>Propriety Number 4:<br /> +<div class="outline-text-5" id="text-orgf4ecdc0"> +<p> +<b>[∀ x ∈ E, P(x)] ∨ [∀ x ∈ E, Q(x)] <i>⇒</i> ∀x ∈ E, [P(x) ∨ Q(x)]</b> +</p> + + +<p> +<i>Same here, implication and NOT en equivalence</i> +</p> +</div> +</li> +</ul> +</div> +</div> +<div id="outline-container-org6421557" class="outline-3"> +<h3 id="org6421557">Multi-parameter proprieties :</h3> +<div class="outline-text-3" id="text-org6421557"> +<p> +A propriety P can depend on two or more parameters, for convenience we call them x,y,z…etc +</p> +</div> +<ul class="org-ul"> +<li><a id="org314cff3"></a>Example :<br /> +<div class="outline-text-6" id="text-org314cff3"> +<p> +P(x,y): x+y > 0 +</p> + + +<p> +P(0,1) is a True proposition +</p> + + +<p> +P(-2,-1) is a False one +</p> +</div> +</li> +<li><a id="orga9b6089"></a>WARNING :<br /> +<div class="outline-text-6" id="text-orga9b6089"> +<p> +∀x ∈ E, ∃y ∈ F , P(x,y) +</p> + + +<p> +∃y ∈ F, ∀x ∈ E , P(x,y) +</p> + + +<p> +Are different because in the first one y depends on x, while in the second one, it doesn’t +</p> +</div> +<ul class="org-ul"> +<li><a id="org3c5e5a8"></a>Example :<br /> +<div class="outline-text-7" id="text-org3c5e5a8"> +<p> +∀ x ∈ ℕ , ∃ y ∈ ℕ y > x -–— True +</p> + + +<p> +∃ y ∈ ℕ , ∀ x ∈ ℕ y > x -–— False +</p> +</div> +</li> +</ul> +</li> +</ul> +<li><a id="org2f3208e"></a>Proprieties :<br /> +<div class="outline-text-5" id="text-org2f3208e"> +<ol class="org-ol"> +<li>not(∀x ∈ E ,∃y ∈ F P(x,y)) ⇔ ∃x ∈ E, ∀y ∈ F not(P(x,y))</li> +<li>not(∃x ∈ E ,∀y ∈ F P(x,y)) ⇔ ∀x ∈ E, ∃y ∈ F not(P(x,y))</li> +</ol> +</div> +</li> +</ul> +</div> +<div id="outline-container-orgc25dd7a" class="outline-3"> +<h3 id="orgc25dd7a">Methods of mathematical reasoning :</h3> +<div class="outline-text-3" id="text-orgc25dd7a"> +</div> +<div id="outline-container-orgf843851" class="outline-4"> +<h4 id="orgf843851">Direct reasoning :</h4> +<div class="outline-text-4" id="text-orgf843851"> +<p> +To show that an implication P ⇒ Q is true, we suppose that P is true and we show that Q is true +</p> +</div> +<ul class="org-ul"> +<li><a id="org6dd2136"></a>Example:<br /> +<div class="outline-text-5" id="text-org6dd2136"> +<p> +Let a,b be two Real numbers, we have to prove that <b>a² + b² = 1 ⇒ |a + b| ≤ 2</b> +</p> + + +<p> +We suppose that a²+b² = 1 and we prove that |a + b| ≤ 2 +</p> + + +<p> +a²+b²=1 ⇒ b² = 1 - a² ; a² = 1 - b² +</p> + + +<p> +a²+b²=1 ⇒ 1 - a² ≥ 0 ; 1 - b² ≥ 0 +</p> + + +<p> +a²+b²=1 ⇒ a² ≤ 1 ; b² ≤ 1 +</p> + + +<p> +a²+b²=1 ⇒ -1 ≤ a ≤ 1 ; -1 ≤ b ≤ 1 +</p> + + +<p> +a²+b²=1 ⇒ -2 ≤ a + b ≤ 2 +</p> + + +<p> +a²+b²=1 ⇒ |a + b| ≤ 2 <b>Which is what we wanted to prove, therefor the implication is correct</b> +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-orga59c0ad" class="outline-4"> +<h4 id="orga59c0ad">Reasoning by the Absurd:</h4> +<div class="outline-text-4" id="text-orga59c0ad"> +<p> +To prove that a proposition is True, we suppose that it’s False and we must come to a contradiction +</p> + + +<p> +And to prove that an implication P ⇒ Q is true using the reasoning by the absurd, we suppose that P ∧ not(Q) is true, and then we come to a contradiction as well +</p> +</div> +<ul class="org-ul"> +<li><a id="orgf8b9f83"></a>Example:<br /> +<div class="outline-text-5" id="text-orgf8b9f83"> +<p> +Prove that this proposition is correct using the reasoning by the absurd : ∀x ∈ ℝ* , sqrt(1+x²) ≠ 1 + x²/2 +</p> + + +<p> +We assume that ∃ x ℝ* , sqrt(1+x²) = 1 + x²/2 +</p> + + +<p> +sqrt(1+x²) = 1 + x²/2 ; 1 + x² = (1+x²/2)² ; 1 + x² = 1 + x^4/4 + x² ; x^(4)/4 = 0 … Which contradicts with our proposition, since x = 4 and we are working on the ℝ* set +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-orgcc285c2" class="outline-4"> +<h4 id="orgcc285c2">Reasoning by contraposition:</h4> +<div class="outline-text-4" id="text-orgcc285c2"> +<p> +If an implication P ⇒ Q is too hard to prove, we just have to prove not(Q) ⇒ not(P) is true !!! or in other words that both not(P) and not(Q) are true +</p> +</div> +</div> +<div id="outline-container-org2e67808" class="outline-4"> +<h4 id="org2e67808">Reasoning by counter example:</h4> +<div class="outline-text-4" id="text-org2e67808"> +<p> +To prove that a proposition ∀x ∈ E, P(x) is false, all we have to do is find a single value of x from E such as not(P(x)) is true +</p> +</div> +</div> +</div> +</div> +<div id="outline-container-org7440601" class="outline-2"> +<h2 id="org7440601">3eme Cours : <i>Oct 9</i></h2> +<div class="outline-text-2" id="text-org7440601"> +</div> +<div id="outline-container-org70aa2db" class="outline-4"> +<h4 id="org70aa2db">Reasoning by recurrence :</h4> +<div class="outline-text-4" id="text-org70aa2db"> +<p> +P is a propriety dependent of <b>n ∈ ℕ</b>. If for n0 ∈ ℕ P(n0) is true, and if for n ≥ n0 (P(n) ⇒ P(n+1)) is true. Then P(n) is true for n ≥ n0 +</p> +</div> +<ul class="org-ul"> +<li><a id="org52e5298"></a>Example:<br /> +<div class="outline-text-5" id="text-org52e5298"> +<p> +Let’s prove that ∀ n ≥ 1 , (n,k=1)Σk = [n(n+1)]/2 +</p> + + +<p> +P(n) : (n,k=1)Σk = [n(n+1)]/2 +</p> + + + +<p> +<b>Pour n = 1:</b> (1,k=1)Σk = 1 ; [n(n+1)]/2 = 1 . <b>So P(1) is true</b> +</p> + + + +<p> +For n ≥ 1. We assume that P(n) is true, OR : <b>(n, k=1)Σk = n(n+1)/2</b>. We now have to prove that P(n+1) is true, Or : <b>(n+1, k=1)Σk = (n+1)(n+2)/2</b> +</p> + + +<p> +(n+1, k=1)Σk = 1 + 2 + …. + n + (n+1) ; (n+1, k=1)Σk = (n, k=1)Σk + (n+1) ; = n(n+1)/2 + (n+1) ; = [n(n+1) + 2(n+1)]/2 ; = <b>[(n+2)(n+1)]/2</b> <i>WHICH IS WHAT WE NEEDED TO FIND</i> +</p> + + +<p> +<b>Conclusion: ∀n ≥ 1 , (n,k=1)Σk = n(n+1)/2</b> +</p> +</div> +</li> +</ul> +</div> +</div> +<div id="outline-container-orga6a518d" class="outline-2"> +<h2 id="orga6a518d">4eme Cours : Chapitre 2 : Sets and Operations</h2> +<div class="outline-text-2" id="text-orga6a518d"> +</div> +<div id="outline-container-org31e3615" class="outline-3"> +<h3 id="org31e3615">Definition of a set :</h3> +<div class="outline-text-3" id="text-org31e3615"> +<p> +A set is a collection of objects that share the sane propriety +</p> +</div> +</div> +<div id="outline-container-orgfa9bfd1" class="outline-3"> +<h3 id="orgfa9bfd1">Belonging, inclusion, and equality :</h3> +<div class="outline-text-3" id="text-orgfa9bfd1"> +<ol class="org-ol"> +<li>Let E be a set. If x is an element of E, we say that x belongs to E we write <b>x ∈ E</b>, and if it doesn’t, we write <b>x ∉ E</b></li> +<li>A set E is included in a set F if all elements of E are elements of F and we write <b>E ⊂ F ⇔ (∀x , x ∈ E ⇒ x ∈ F)</b>. We say that E is a subset of F, or a part of F. The negation of this propriety is : <b>E ⊄ F ⇔ ∃x , x ∈ E and x ⊄ F</b></li> +<li>E and F are equal if E is included in F and F is included in E, and we write <b>E = F ⇔ (E ⊂ F) et (F ⊂ E)</b></li> +<li>The empty set (symbolized by ∅) is a set without elements, and is included in all sets (by convention) : <b>∅ ⊂ E</b></li> +</ol> +</div> +</div> +<div id="outline-container-org2a19707" class="outline-3"> +<h3 id="org2a19707">Intersections and reunions :</h3> +<div class="outline-text-3" id="text-org2a19707"> +</div> +<div id="outline-container-org6a5f566" class="outline-4"> +<h4 id="org6a5f566">Intersection:</h4> +<div class="outline-text-4" id="text-org6a5f566"> +<p> +E ∩ F = {x / x ∈ E AND x ∈ F} ; x ∈ E ∩ F ⇔ x ∈ F AND x ∈ F +</p> + + +<p> +x ∉ E ∩ F ⇔ x ∉ E OR x ∉ F +</p> +</div> +</div> +<div id="outline-container-org9bc9aeb" class="outline-4"> +<h4 id="org9bc9aeb">Union:</h4> +<div class="outline-text-4" id="text-org9bc9aeb"> +<p> +E ∪ F = {x / x ∈ E OR x ∈ F} ; x ∈ E ∪ F ⇔ x ∈ F OR x ∈ F +</p> + + +<p> +x ∉ E ∪ F ⇔ x ∉ E AND x ∉ F +</p> +</div> +</div> +<div id="outline-container-org9a7f719" class="outline-4"> +<h4 id="org9a7f719">Difference between two sets:</h4> +<div class="outline-text-4" id="text-org9a7f719"> +<p> +E\F(Which is also written as : E - F) = {x / x ∈ E and x ∉ F} +</p> +</div> +</div> +<div id="outline-container-org5f5c721" class="outline-4"> +<h4 id="org5f5c721">Complimentary set:</h4> +<div class="outline-text-4" id="text-org5f5c721"> +<p> +If F ⊂ E. E - F is the complimentary of F in E. +</p> + + +<p> +FCE = {x /x ∈ E AND x ∉ F} <b>ONLY WHEN F IS A SUBSET OF E</b> +</p> +</div> +</div> +<div id="outline-container-orga285d1d" class="outline-4"> +<h4 id="orga285d1d">Symentrical difference</h4> +<div class="outline-text-4" id="text-orga285d1d"> +<p> +E Δ F = (E - F) ∪ (F - E) ; = (E ∪ F) - (E ∩ F) +</p> +</div> +</div> +</div> +<div id="outline-container-orgc12e73b" class="outline-3"> +<h3 id="orgc12e73b">Proprieties :</h3> +<div class="outline-text-3" id="text-orgc12e73b"> +<p> +Let E,F and G be 3 sets. We have : +</p> +</div> +<div id="outline-container-org31d8697" class="outline-4"> +<h4 id="org31d8697">Commutativity:</h4> +<div class="outline-text-4" id="text-org31d8697"> +<p> +E ∩ F = F ∩ E +E ∪ F = F ∪ E +</p> +</div> +</div> +<div id="outline-container-org7080d99" class="outline-4"> +<h4 id="org7080d99">Associativity:</h4> +<div class="outline-text-4" id="text-org7080d99"> +<p> +E ∩ (F ∩ G) = (E ∩ F) ∩ G +E ∪ (F ∪ G) = (E ∪ F) ∪ G +</p> +</div> +</div> +<div id="outline-container-org13da04d" class="outline-4"> +<h4 id="org13da04d">Distributivity:</h4> +<div class="outline-text-4" id="text-org13da04d"> +<p> +E ∩ (F ∪ G) = (E ∩ F) ∪ (E ∩ G) +E ∪ (F ∩ G) = (E ∪ F) ∩ (E ∪ G) +</p> +</div> +</div> +<div id="outline-container-orgaa33b71" class="outline-4"> +<h4 id="orgaa33b71">Lois de Morgan:</h4> +<div class="outline-text-4" id="text-orgaa33b71"> +<p> +If E ⊂ G and F ⊂ G ; +</p> + +<p> +(E ∩ F)CG = ECG ∪ FCG ; (E ∪ F)CG = ECG ∩ FCG +</p> +</div> +</div> +<div id="outline-container-org7e7db42" class="outline-4"> +<h4 id="org7e7db42">An other one:</h4> +<div class="outline-text-4" id="text-org7e7db42"> +<p> +E - (F ∩ G) = (E-F) ∪ (E-G) ; E - (F ∪ G) = (E-F) ∩ (E-G) +</p> +</div> +</div> +<div id="outline-container-orgd02bd7f" class="outline-4"> +<h4 id="orgd02bd7f">An other one:</h4> +<div class="outline-text-4" id="text-orgd02bd7f"> +<p> +E ∩ ∅ = ∅ ; E ∪ ∅ = E +</p> +</div> +</div> +<div id="outline-container-org99eb39a" class="outline-4"> +<h4 id="org99eb39a">And an other one:</h4> +<div class="outline-text-4" id="text-org99eb39a"> +<p> +E ∩ (F Δ G) = (E ∩ F) Δ (E ∩ G) +</p> +</div> +</div> +<div id="outline-container-org3e9b2ef" class="outline-4"> +<h4 id="org3e9b2ef">And the last one:</h4> +<div class="outline-text-4" id="text-org3e9b2ef"> +<p> +E Δ ∅ = E ; E Δ E = ∅ +</p> +</div> +</div> +</div> +</div> +</div> +<div id="postamble" class="status"> +<p class="author">Author: Crystal</p> +<p class="date">Created: 2023-10-11 Wed 19:04</p> +</div> +</body> +</html> \ No newline at end of file diff --git a/uni_notes/alsd.html b/uni_notes/alsd.html new file mode 100755 index 0000000..9b0fa1c --- /dev/null +++ b/uni_notes/alsd.html @@ -0,0 +1,84 @@ +<?xml version="1.0" encoding="utf-8"?> +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" +"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> +<head> +<!-- 2023-10-01 Sun 20:23 --> +<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> +<meta name="viewport" content="width=device-width, initial-scale=1" /> +<title>ALSD1</title> +<meta name="author" content="Crystal" /> +<meta name="generator" content="Org Mode" /> +<link rel="stylesheet" type="text/css" href="../src/css/colors.css"/> +<link rel="stylesheet" type="text/css" href="../src/css/style.css"/> +</head> +<body> +<div id="content" class="content"> +<h1 class="title">ALSD1</h1> + +<div id="outline-container-org2b6a11d" class="outline-2"> +<h2 id="org2b6a11d">Contenu de la Matiére</h2> +<div class="outline-text-2" id="text-org2b6a11d"> +</div> +<div id="outline-container-org5340dec" class="outline-3"> +<h3 id="org5340dec">Chapitre 1: Elements de Base</h3> +<div class="outline-text-3" id="text-org5340dec"> +<ul class="org-ul"> +<li>Algorithmique, procésseur, action.</li> +<li>Programme et languages de programmation.</li> +<li>Analyse des problémes.</li> +</ul> +</div> +</div> + +<div id="outline-container-org92c03d8" class="outline-3"> +<h3 id="org92c03d8">Chapitre 2: Présentation du formalisme Algorithmique</h3> +</div> +<div id="outline-container-orgde3a680" class="outline-3"> +<h3 id="orgde3a680">Chapitre 3: Eléments de base du language C</h3> +</div> +<div id="outline-container-org4eb24e1" class="outline-3"> +<h3 id="org4eb24e1">Chapitre 4: Modularité( Fonction et Procédure )</h3> +</div> +<div id="outline-container-org6a31b85" class="outline-3"> +<h3 id="org6a31b85">Chapitre 5: Les structures des données statiques</h3> +</div> +</div> + + +<div id="outline-container-org7ccc87b" class="outline-2"> +<h2 id="org7ccc87b">Premier cours : Algorithmes <i>Oct 1</i> :</h2> +<div class="outline-text-2" id="text-org7ccc87b"> +</div> +<div id="outline-container-orge430c1a" class="outline-3"> +<h3 id="orge430c1a">Définition d’un algorithm :</h3> +<div class="outline-text-3" id="text-orge430c1a"> +<p> +Un ensemble d’opérations ecrites dans le language naturel. +</p> +</div> + +<div id="outline-container-orgbaecdfc" class="outline-4"> +<h4 id="orgbaecdfc">Example d’un Algo : Résolution d’une équation du second ordre (ax²+bx+c=0)</h4> +<div class="outline-text-4" id="text-orgbaecdfc"> +<ol class="org-ol"> +<li>Si a=0 ET b=0 alors <b>l’équation n’est pas du 2nd ordre</b>.</li> +<li>Si a=0 et b≠0 alors <b>x= -c/5</b> .</li> +<li>Si a≠0 alors <b>calculer Δ= b²-4ac</b> : +<ol class="org-ol"> +<li>Si Δ=0 alors <b>x=-b/2a</b>.</li> +<li>Si Δ<0 alors <b>l’équation n’as pas de solution</b>.</li> +<li>Si Δ>0 alors <b>x=[-b±sqrt(Δ)]/2a</b></li> +</ol></li> +</ol> +</div> +</div> +</div> +</div> +</div> +<div id="postamble" class="status"> +<p class="author">Author: Crystal</p> +<p class="date">Created: 2023-10-01 Sun 20:23</p> +</div> +</body> +</html> \ No newline at end of file diff --git a/uni_notes/analyse.html b/uni_notes/analyse.html new file mode 100755 index 0000000..20d8972 --- /dev/null +++ b/uni_notes/analyse.html @@ -0,0 +1,1123 @@ +<?xml version="1.0" encoding="utf-8"?> +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" +"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> +<head> +<!-- 2023-10-11 Wed 19:18 --> +<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> +<meta name="viewport" content="width=device-width, initial-scale=1" /> +<title>Analyse 1</title> +<meta name="author" content="Crystal" /> +<meta name="generator" content="Org Mode" /> +<link rel="stylesheet" type="text/css" href="../src/css/colors.css"/> +<link rel="stylesheet" type="text/css" href="../src/css/style.css"/> +</head> +<body> +<div id="content" class="content"> +<h1 class="title">Analyse 1</h1> +<div id="outline-container-org32ad572" class="outline-2"> +<h2 id="org32ad572">Contenu de la Matiére</h2> +<div class="outline-text-2" id="text-org32ad572"> +</div> +<div id="outline-container-org156647d" class="outline-3"> +<h3 id="org156647d">Chapitre 1 : Quelque propriétés de ℝ</h3> +<div class="outline-text-3" id="text-org156647d"> +<ul class="org-ul"> +<li>Structure algébrique de ℝ</li> +<li>L’ordre dans ℝ</li> +<li>Majorant, minorant, borne superieure, borne inférieure</li> +</ul> +</div> +</div> +<div id="outline-container-org2064013" class="outline-3"> +<h3 id="org2064013">Chapitre 2 : Les suites numériques réelles</h3> +<div class="outline-text-3" id="text-org2064013"> +<ul class="org-ul"> +<li>Définition : convergence, opérations sur les suites convergentes</li> +<li>Theoréme de convergence, Theoréme de <span class="underline">_</span> suites, sans suites, extension au limites infinies</li> +<li>Suites de cauchy, suites adjacentes et suites récurentes</li> +</ul> +</div> +</div> +<div id="outline-container-org74215b0" class="outline-3"> +<h3 id="org74215b0">Chapitre 3 : Limites et continuité des fonctions réelles d’une variable réelle</h3> +<div class="outline-text-3" id="text-org74215b0"> +<ul class="org-ul"> +<li>Les limites : définition, opérations sur les limites, les formes inditerminées</li> +<li>La continuité : définition, Theorémes fondamentaux</li> +<li>La continuité informe les fonctions Lepchitziennes</li> +</ul> +</div> +</div> +<div id="outline-container-orgfad0512" class="outline-3"> +<h3 id="orgfad0512">Chapitre 4 : La dérivabilité et son interprétation géometrique</h3> +<div class="outline-text-3" id="text-orgfad0512"> +<ul class="org-ul"> +<li>Opérations sur les fonctions dérivales, Theoréme de Rolle, Theoréme des accroissements finis, régle de L’Hopital et formule de Taylor</li> +</ul> +</div> +</div> +<div id="outline-container-org2c42f3a" class="outline-3"> +<h3 id="org2c42f3a">Chapitre 5 : Les fonctions trigonométriques réciproques, fonctions hypérboliques réciproques</h3> +<div class="outline-text-3" id="text-org2c42f3a"> +<ul class="org-ul"> +<li>Comparaison asymptotique</li> +<li>Symbole de lamdau (lambda ?), et notions des fonctions équivalentes</li> +<li>Développements limites polynominaux (D.L) et opérations sur les D.L</li> +<li>Généralisations des D.L</li> +<li>Application au calcul de limite et l’étude des branches infinies</li> +</ul> +</div> +</div> +</div> +<div id="outline-container-org4aeab0e" class="outline-2"> +<h2 id="org4aeab0e">Premier cours : Quelque propriétés de ℝ <i>Sep 26</i> :</h2> +<div class="outline-text-2" id="text-org4aeab0e"> +</div> +<div id="outline-container-org1c15646" class="outline-3"> +<h3 id="org1c15646">La loi de composition interne dans E :</h3> +<div class="outline-text-3" id="text-org1c15646"> +<p> +@ : E x E —> E + (x,y) —> x @ y +</p> + +<p> +@ est une lois de composition interne seulement si : +</p> + +<p> +<b>∀ x,y ε E</b> +</p> +</div> +<div id="outline-container-orge4f4285" class="outline-4"> +<h4 id="orge4f4285"><b>Example : Addition</b></h4> +<div class="outline-text-4" id="text-orge4f4285"> +<p> +Est ce que l’addition (+) est L.C.I dans ℕ ? +</p> + +<p> +ℕ x ℕ —> ℕ +</p> + +<p> +(x,y) —> x + y ? <i>En gros : Pour que l’addition soit une L.C.I dans ℕ, il faut que: quand on additionne <b>n’importe quel</b> chiffre x et y de N, il faut que le résultat appertiens aussi a ℕ</i> +</p> + +<p> +∀ x,y ∈ ℕ , x + y ∈ ℕ <i>En gros: Pour TOUTE valeur de x et y appartenant a ℕ, leur somme est toujours dans ℕ</i> +</p> + +<p> +Donc : + est L.C.I dans ℕ +</p> +</div> +</div> +<div id="outline-container-orgf679b38" class="outline-4"> +<h4 id="orgf679b38"><b>Example : soustraction</b></h4> +<div class="outline-text-4" id="text-orgf679b38"> +<p> +Est ce que la soustraction (-) est L.C.I dans ℕ? +</p> + +<p> +ℕ x ℕ —> ℕ +</p> + +<p> +(x,y) —> x - y ? +</p> + + +<p> +∃ x , y ∈ ℕ , x - y ∉ ℕ <i>En gros: il existe au moins une valeur de x et y dans ℕ tel que leur différence n’est <b>PAS</b> dans ℕ . tel que : si x est 5, et y c’est 9. Leur différence est -4, qui appartiens pas a ℕ</i> +</p> +</div> +</div> +</div> +<div id="outline-container-org5bd802e" class="outline-3"> +<h3 id="org5bd802e">La loi de composition externe dans E :</h3> +<div class="outline-text-3" id="text-org5bd802e"> +<p> +@ est L.C.E dans E, K est un corps +</p> + +<p> +K x E —> E +</p> + +<p> +(a,x) —> a @ x +</p> + +<p> +∀ (a , x) ∈ K x E , a @ x ∈ E +</p> +</div> +</div> +<div id="outline-container-orgbd9ed54" class="outline-3"> +<h3 id="orgbd9ed54">Groupes :</h3> +<div class="outline-text-3" id="text-orgbd9ed54"> +<p> +<i>Soit E un ensemble, soit @ une L.C.I dans E</i> +</p> + +<p> +(E, @) est un groupe Si : +</p> +</div> +<div id="outline-container-org8a61f65" class="outline-4"> +<h4 id="org8a61f65">Il contiens un élement neutre</h4> +<div class="outline-text-4" id="text-org8a61f65"> +<p> +∀ x ∈ E ; ∃ e ∈ E +</p> + +<p> +x @ e = e @ x = x +</p> + +<p> +On appelle <b>e</b> élement neutre +</p> + +<p> +<i>Ex: (ℕ,+) accepte un élement neutre, qui est 0, parceque x + 0 = 0 + x = x….cependent (ℕ,+) n’est pas un groupe. La raison est dans la prochaine condition</i> +</p> +</div> +</div> +<div id="outline-container-orgd7a2bc2" class="outline-4"> +<h4 id="orgd7a2bc2">Il contiens un élément symétrique</h4> +<div class="outline-text-4" id="text-orgd7a2bc2"> +<p> +∀ x ∈ E ; ∃ x’ ∈ E ; x @ x’ = x’ @ x = e +</p> + +<p> +On appelle <b>x’</b> élèment symétrique +</p> + +<p> +<i>Dans l’example en haut, on remarque qu’il n’y ya pas de chiffre x’ pour chaque chiffre x, qui est, l’hors de leur addition est egal a e (0), tout simplement car:</i> +</p> + +<p> +<i>x + x’ = e ; x + x’ = 0 ; x = -x’</i> +</p> + +<p> +<b>Or, Dans ℕ, on a pas de nombres négatifs</b> +</p> +</div> +</div> +<div id="outline-container-org942f964" class="outline-4"> +<h4 id="org942f964">@ est cummutative :</h4> +<div class="outline-text-4" id="text-org942f964"> +<p> +∀ (x , x’) ∈ E x E ; x @ x’ = x’ @ x +</p> + +<p> +<i>L’addition est cummutative, la soustraction ne l’es pas. 5 + 3 ou 3 + 5 est pareil, mais 5 - 3 et 3 - 5 sont différents</i> +</p> +</div> +</div> +</div> +<div id="outline-container-orgfd870b2" class="outline-3"> +<h3 id="orgfd870b2">Anneaux :</h3> +<div class="outline-text-3" id="text-orgfd870b2"> +<p> +Soit E un ensemble, (E , @ , !) est un anneau si : +</p> +</div> +<div id="outline-container-orgb630ae9" class="outline-4"> +<h4 id="orgb630ae9">(E ; @) est un groupe cummutatif</h4> +</div> +<div id="outline-container-org83da3f3" class="outline-4"> +<h4 id="org83da3f3">! est une loi associative :</h4> +<div class="outline-text-4" id="text-org83da3f3"> +<p> +∀ x , y , z ∈ E +</p> + +<p> +(x ! y) ! z = x ! (y ! z) +</p> +</div> +</div> +<div id="outline-container-org228d644" class="outline-4"> +<h4 id="org228d644">Distribution de ! par rapport à @ :</h4> +<div class="outline-text-4" id="text-org228d644"> +<p> +∀ x , y , z ∈ E +</p> + +<p> +(x @ y) ! z = ( x ! z ) @ ( y ! z ) +</p> +</div> +</div> +<div id="outline-container-org188625a" class="outline-4"> +<h4 id="org188625a">L’existance d’un élèment neutre de ! :</h4> +<div class="outline-text-4" id="text-org188625a"> +<p> +∀ x ∈ E , ∃ e ∈ E , x ! e = e ! x = x +</p> +</div> +</div> +<div id="outline-container-orge450b40" class="outline-4"> +<h4 id="orge450b40">! est cummutative :</h4> +<div class="outline-text-4" id="text-orge450b40"> +<p> +∀ x , y ∈ E , x ! y = y ! x +</p> +</div> +</div> +</div> +<div id="outline-container-orgb215ac1" class="outline-3"> +<h3 id="orgb215ac1">Corps :</h3> +<div class="outline-text-3" id="text-orgb215ac1"> +<p> +(E , @ , !) est un corps si les 5 conditions en haut sont vérifiées + cette condition : +</p> +</div> +<div id="outline-container-org51d906d" class="outline-4"> +<h4 id="org51d906d">La symétrie :</h4> +<div class="outline-text-4" id="text-org51d906d"> +<p> +∀ x ∈ E ; ∃ x’ ∈ E , x ! x’ = x’ ! x = e +</p> + +<p> +x’ est l’élément symétrique de x par rapport à ! +(sauf élément neutre première lois ) +</p> +</div> +</div> +</div> +<div id="outline-container-orgea20262" class="outline-3"> +<h3 id="orgea20262">Exercice : (ℝ, +, x) corps ou pas ?</h3> +<div class="outline-text-3" id="text-orgea20262"> +</div> +<div id="outline-container-org63b1ea5" class="outline-4"> +<h4 id="org63b1ea5">Est-ce un Anneau ?</h4> +<div class="outline-text-4" id="text-org63b1ea5"> +<ul class="org-ul"> +<li>(ℝ, +) est un groupe commutatif</li> +<li>x est une loi associative : (a x b) x c = a x (b x c)</li> +<li>On peut distribuer x par rapport a + : (a + b) x c = (a x c) + (b x c)</li> +<li>Il existe un élément neutre de x which is 1 : a x 1 = 1 x a = a</li> +<li>La multiplication est commutative : a x b = b x a</li> +</ul> + +<p> +Oui c’est un anneau +</p> +</div> +</div> +<div id="outline-container-org1c18b97" class="outline-4"> +<h4 id="org1c18b97">Est-ce un corps ?</h4> +<div class="outline-text-4" id="text-org1c18b97"> +<ul class="org-ul"> +<li>Oui : ∀ x ∈ ℝ\{e} ; x * x’ = 1</li> +</ul> +</div> +</div> +</div> +</div> +<div id="outline-container-org86b97eb" class="outline-2"> +<h2 id="org86b97eb">2nd cours :L’ordre dans ℝ, Majorant, minorant, borne superieure, borne inférieure <i>Oct 3</i> :</h2> +<div class="outline-text-2" id="text-org86b97eb"> +</div> +<div id="outline-container-org1f1c4d8" class="outline-3"> +<h3 id="org1f1c4d8">L’ordre dans ℝ</h3> +<div class="outline-text-3" id="text-org1f1c4d8"> +<p> +(ℝ, +, x) est un corps, Soit R une relation d’ordre dans ℝ si : +</p> + +<ol class="org-ol"> +<li><p> +R est antisymétrique : +</p> + +<p> +∀ x, y ℝ ; (x R y et y R x) ⇒ (x = y) +</p></li> + +<li><p> +R est reflexive : +</p> + +<p> +∀ x ∈ ℝ ; x R x +</p></li> + +<li>R est transitive : +∀ x, y, z ∈ ℝ , (x R y and y R z) ⇒ x R z</li> +</ol> +</div> +<div id="outline-container-orgc178857" class="outline-4"> +<h4 id="orgc178857">Exemples :</h4> +<div class="outline-text-4" id="text-orgc178857"> +</div> +<ul class="org-ul"> +<li><a id="org7b5f181"></a>Exemple numéro 1:<br /> +<div class="outline-text-5" id="text-org7b5f181"> +<p> +(ℝ , +, x) est un corps. Est ce la relation < est une relation d’ordre dans ℝ ? +</p> + + +<p> +Non, pourquoi ? parce que elle est pas réflexive : ∀ x ∈ ℝ, x < x <b><b>is obviously false</b></b> +</p> +</div> +</li> +<li><a id="org7f902ab"></a>Exemple numéro 2:<br /> +<div class="outline-text-5" id="text-org7f902ab"> +<p> +(ℝ , +, x) est un corps. Est ce la relation ≥ est une relation d’ordre dans ℝ ? +</p> + +<ol class="org-ol"> +<li>(Antisymétrique) ∀ x, y ℝ ; (x ≥ y AND y ≥ x) ⇒ x = y is true</li> +<li>(Réflexive) ∀ x, y ℝ ; x ≥ x is true</li> +<li>(Transitive) ∀ x, y, z ℝ ; (x ≥ y AND y ≥ z) ⇒ x ≥ z is also true</li> +</ol> +</div> +</li> +</ul> +</div> +</div> +<div id="outline-container-org446671f" class="outline-3"> +<h3 id="org446671f">Majorant, minorant, borne supérieure, borne inférieure</h3> +<div class="outline-text-3" id="text-org446671f"> +</div> +<div id="outline-container-org86c077d" class="outline-4"> +<h4 id="org86c077d">Majorant:</h4> +<div class="outline-text-4" id="text-org86c077d"> +<p> +Soit E un sous-ensemble de ℝ (E ⊆ ℝ) +</p> + + +<p> +Soit a ∈ ℝ, a est un majorant de E Si :∀ x ∈ E , x ≤ a +</p> +</div> +</div> +<div id="outline-container-org200a2c3" class="outline-4"> +<h4 id="org200a2c3">Minorant:</h4> +<div class="outline-text-4" id="text-org200a2c3"> +<p> +Soit E un sous-ensemble de ℝ (E ⊆ ℝ) +</p> + + +<p> +Soit b ∈ ℝ, b est un minorant de E Si :∀ x ∈ E , x ≥ b +</p> +</div> +</div> +<div id="outline-container-org46cb24b" class="outline-4"> +<h4 id="org46cb24b">Borne supérieure:</h4> +<div class="outline-text-4" id="text-org46cb24b"> +<p> +La borne supérieure est le plus petit des majorants <i>Sup(E) = Borne supérieure</i> +</p> +</div> +</div> +<div id="outline-container-orgbb1980a" class="outline-4"> +<h4 id="orgbb1980a">Borne inférieure:</h4> +<div class="outline-text-4" id="text-orgbb1980a"> +<p> +La borne inférieure est le plus grand des minorant <i>Inf(E) = Borne inférieure</i> +</p> +</div> +</div> +<div id="outline-container-org3159ba9" class="outline-4"> +<h4 id="org3159ba9">Maximum :</h4> +<div class="outline-text-4" id="text-org3159ba9"> +<p> +E ⊆ ℝ, a est un maximum de E (Max(E)) Si : a ∈ E ; ∀x ∈ E, x ≤ a. +</p> +</div> +</div> +<div id="outline-container-orgf43e25a" class="outline-4"> +<h4 id="orgf43e25a">Minimum :</h4> +<div class="outline-text-4" id="text-orgf43e25a"> +<p> +E ⊆ ℝ, b est un minimum de E (Min(E)) Si : b ∈ E ; ∀x ∈ E, x ≥ b. +</p> +</div> +</div> +<div id="outline-container-orga08da55" class="outline-4"> +<h4 id="orga08da55">Remarques :</h4> +<div class="outline-text-4" id="text-orga08da55"> +<p> +A et B deux ensembles bornés (Minoré et Majoré) : +</p> +<ol class="org-ol"> +<li>A ∪ B est borné</li> +<li>A ∩ B est borné</li> +<li>Sup(A ∪ B)= Max(sup A, sup B)</li> +<li>Inf(A ∩ B)= Min(inf A, inf B)</li> +<li>Sup(A ∩ B)= Min(sup A, sup B) <i>Le plus petit des Supérieur de A et B</i></li> +<li>Inf(A ∩ B)= Max(inf A, inf B) <i>Le plus grand des inférieur de A et B</i></li> +</ol> +</div> +</div> +</div> +</div> +<div id="outline-container-org05e73a3" class="outline-2"> +<h2 id="org05e73a3">3rd cours :Les suites numériques <i>Oct 5</i> :</h2> +<div class="outline-text-2" id="text-org05e73a3"> +</div> +<div id="outline-container-orgcd4347a" class="outline-4"> +<h4 id="orgcd4347a">Définition :</h4> +<div class="outline-text-4" id="text-orgcd4347a"> +<p> +Soit (Un)n ∈ ℕ une suite numérique , (Un)n est une application de ℕ dans ℝ: +</p> + + +<p> +ℕ -—> ℝ +</p> + + +<p> +n -—> U(n) = Un +</p> + +<ol class="org-ol"> +<li>(Un) ou (Un)n ∈ ℝ : une suite</li> +<li>Un : terme général</li> +</ol> +</div> +<ul class="org-ul"> +<li><a id="org495644f"></a>Exemple :<br /> +<div class="outline-text-6" id="text-org495644f"> +<p> +U : ℕ* -—> ℝ +</p> + + +<p> +n -—> 1/n +</p> + + +<p> +(Un) est une suite définit par Un = 1/n +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-orgdf8ea2e" class="outline-4"> +<h4 id="orgdf8ea2e">Définition N°2 :</h4> +<div class="outline-text-4" id="text-orgdf8ea2e"> +<p> +On peut définir une suite â partir d’une relation de récurrence entre deux termes successifs et le premier terme. +</p> +</div> +<ul class="org-ul"> +<li><a id="org0c7274e"></a>Exemple :<br /> +<div class="outline-text-6" id="text-org0c7274e"> +<p> +U(n+1) = Un /2 +</p> + + +<p> +U(1)= 1 +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-orgecfb02c" class="outline-3"> +<h3 id="orgecfb02c">Opérations sur les suites :</h3> +<div class="outline-text-3" id="text-orgecfb02c"> +</div> +<div id="outline-container-orgc56ccde" class="outline-4"> +<h4 id="orgc56ccde">La somme :</h4> +<div class="outline-text-4" id="text-orgc56ccde"> +<p> +Soient (Un) et (Vn) deux suites, la somme de (Un) et (Vn) est une suite de terme général Un + Vn +</p> +</div> +</div> +<div id="outline-container-orgfe0c8af" class="outline-4"> +<h4 id="orgfe0c8af">Le produit :</h4> +<div class="outline-text-4" id="text-orgfe0c8af"> +<p> +Soient (Un)n et (Vn)n deux suites alors (Un) x (Vn) est une autre suite de terme général Un x Vn +</p> +</div> +</div> +<div id="outline-container-orgfa9fe65" class="outline-4"> +<h4 id="orgfa9fe65">Inverse d’une suite :</h4> +<div class="outline-text-4" id="text-orgfa9fe65"> +<p> +Soit Un une suite de terme général Un alors l’inverse de (Un) est une autre suite (Vn) = 1/(Un) de terme général de Vn = 1/Un +</p> +</div> +</div> +<div id="outline-container-org09108ca" class="outline-4"> +<h4 id="org09108ca">Produit d’une suite par un scalaire :</h4> +<div class="outline-text-4" id="text-org09108ca"> +<p> +Soit (Un) une suite de T.G Un +</p> + + +<p> +∀ λ ∈ ℝ , λ(Un) n ∈ ℕ est une suite de T.G Vn= λUn +</p> +</div> +</div> +</div> +<div id="outline-container-org0d16671" class="outline-3"> +<h3 id="org0d16671">Suite bornée :</h3> +<div class="outline-text-3" id="text-org0d16671"> +<p> +Une suite (Un) est bornée si (Un) majorée et minorée +</p> +</div> +</div> +<div id="outline-container-org1819f18" class="outline-3"> +<h3 id="org1819f18">Suite majorée :</h3> +<div class="outline-text-3" id="text-org1819f18"> +<p> +Soit (Un) une suite +</p> + + +<p> +U : (Un) est majorée par M ∈ ℝ ; ∀ n ∈ ℕ ; ∃ M ∈ ℝ , Un ≤ M +</p> +</div> +</div> +<div id="outline-container-org3c0fbec" class="outline-3"> +<h3 id="org3c0fbec">Suite minorée :</h3> +<div class="outline-text-3" id="text-org3c0fbec"> +<p> +Soit (Un) une suite +</p> + + +<p> +U : (Un) est minorée par M ∈ ℝ ; ∀ n ∈ ℕ ; ∃ M ∈ ℝ , Un ≥ M +</p> +</div> +</div> +<div id="outline-container-orgebf3d3b" class="outline-3"> +<h3 id="orgebf3d3b">Suites monotones :</h3> +<div class="outline-text-3" id="text-orgebf3d3b"> +</div> +<div id="outline-container-org193a450" class="outline-4"> +<h4 id="org193a450">Les suites croissantes :</h4> +<div class="outline-text-4" id="text-org193a450"> +<p> +Soit (Un)n est une suite +</p> + + +<p> +(Un) est croissante si : ∀ n ∈ ℕ ; U(n+1) - Un ≥ 0 ⇔ Un+1 ≥ Un +</p> +</div> +</div> +<div id="outline-container-orgff47924" class="outline-4"> +<h4 id="orgff47924">Les suites décroissantes :</h4> +<div class="outline-text-4" id="text-orgff47924"> +<p> +Soit (Un)n est une suite +</p> + + +<p> +(Un) est décroissante si : ∀ n ∈ ℕ ; U(n+1) - Un ≤ 0 ⇔ Un+1 ≤ Un +</p> +</div> +</div> +</div> +</div> +<div id="outline-container-orgf08c70b" class="outline-2"> +<h2 id="orgf08c70b">Série TD N°1 : <i>Oct 6</i></h2> +<div class="outline-text-2" id="text-orgf08c70b"> +</div> +<div id="outline-container-org24ad469" class="outline-3"> +<h3 id="org24ad469">Exo 1 :</h3> +<div class="outline-text-3" id="text-org24ad469"> +</div> +<div id="outline-container-org542ddd3" class="outline-4"> +<h4 id="org542ddd3">Ensemble A :</h4> +<div class="outline-text-4" id="text-org542ddd3"> +<p> +A = {-1/n , n ∈ ℕ *} +</p> +</div> +<ul class="org-ul"> +<li><a id="org8b8384f"></a>Borne inférieure<br /> +<div class="outline-text-5" id="text-org8b8384f"> +<p> +∀ n ∈ ℕ* , -1/n ≥ -1 . -1 est la borne inférieure de l’ensemble A +</p> +</div> +</li> +<li><a id="orgd7452d9"></a>Minimum :<br /> +<div class="outline-text-5" id="text-orgd7452d9"> +<p> +∀ n ∈ ℕ* , -1/n ≥ -1 . -1 est le Minimum de l’ensemble A +</p> +</div> +</li> +<li><a id="org60371ac"></a>Borne supérieure :<br /> +<div class="outline-text-5" id="text-org60371ac"> +<p> +∀ n ∈ ℕ* , -1/n ≤ 0 . 0 est la borne supérieure de l’ensemble A +</p> +</div> +</li> +<li><a id="orge3e4c79"></a>Maximum :<br /> +<div class="outline-text-5" id="text-orge3e4c79"> +<p> +L’ensemble A n’as pas de maximum +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-org71dc196" class="outline-4"> +<h4 id="org71dc196">Ensemble B :</h4> +<div class="outline-text-4" id="text-org71dc196"> +<p> +B = [-1 , 3[ ∩ ℚ +</p> +</div> +<ul class="org-ul"> +<li><a id="orgb1a63f2"></a>Borne inférieure :<br /> +<div class="outline-text-5" id="text-orgb1a63f2"> +<p> +Inf(B) = Max(inf([-1 , 3[) , inf(ℚ)) +</p> + + +<p> +Puisse que ℚ n’as pas de Borne inférieure, donc par convention c’est <b>-∞</b>, +</p> + + +<p> +<b>Inf(B) = -1</b> +</p> +</div> +</li> +<li><a id="org777951c"></a>Borne supérieure :<br /> +<div class="outline-text-5" id="text-org777951c"> +<p> +Sup(B) = Min(sup([-1 ,3[) , sup(ℚ)) +</p> + + +<p> +Puisse que ℚ n’as pas de Borne supérieure, donc par convention c’est <b>+∞</b>, +</p> + + +<p> +<b>Sup(B) = 3</b> +</p> +</div> +</li> +<li><a id="org8f243eb"></a>Minimum :<br /> +<div class="outline-text-5" id="text-org8f243eb"> +<p> +<b>Min(B) = -1</b> +</p> +</div> +</li> +<li><a id="org7e50683"></a>Maximum :<br /> +<div class="outline-text-5" id="text-org7e50683"> +<p> +L’ensemble B n’as pas de Maximum +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-org22ac863" class="outline-4"> +<h4 id="org22ac863">Ensemble C :</h4> +<div class="outline-text-4" id="text-org22ac863"> +<p> +C = {3n ,n ∈ ℕ} +</p> +</div> +<ul class="org-ul"> +<li><a id="org21834f8"></a>Borne inférieure :<br /> +<div class="outline-text-5" id="text-org21834f8"> +<p> +Inf(C) = 0 +</p> +</div> +</li> +<li><a id="org9559808"></a>Borne supérieure :<br /> +<div class="outline-text-5" id="text-org9559808"> +<p> +Sup(C) = +∞ +</p> +</div> +</li> +<li><a id="org4a57e53"></a>Minimum :<br /> +<div class="outline-text-5" id="text-org4a57e53"> +<p> +Min(C) = 0 +</p> +</div> +</li> +<li><a id="org621b5ba"></a>Maximum :<br /> +<div class="outline-text-5" id="text-org621b5ba"> +<p> +L’ensemble C n’as pas de Maximum +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-orgfbcec21" class="outline-4"> +<h4 id="orgfbcec21">Ensemble D :</h4> +<div class="outline-text-4" id="text-orgfbcec21"> +<p> +D = {1 - 1/n , n ∈ ℕ*} +</p> +</div> +<ul class="org-ul"> +<li><a id="orga8dbb3d"></a>Borne inférieure :<br /> +<div class="outline-text-5" id="text-orga8dbb3d"> +<p> +Inf(D)= 0 +</p> +</div> +</li> +<li><a id="org17babcd"></a>Borne supérieure :<br /> +<div class="outline-text-5" id="text-org17babcd"> +<p> +Sup(D)= 1 +</p> +</div> +</li> +<li><a id="orgbd6de63"></a>Minimum :<br /> +<div class="outline-text-5" id="text-orgbd6de63"> +<p> +Min(D)= 0 +</p> +</div> +</li> +<li><a id="org8c6ce24"></a>Maximum :<br /> +<div class="outline-text-5" id="text-org8c6ce24"> +<p> +L’ensemble D n’as pas de Maximum +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-orgffb8405" class="outline-4"> +<h4 id="orgffb8405">Ensemble E :</h4> +<div class="outline-text-4" id="text-orgffb8405"> +<p> +E = { [2n + (-1)^n]/ n + 1 , n ∈ ℕ } +</p> + + +<p> +<b>Les valeurs que E peut prendre sont : “(2n + 1)/(n+1)” Si n est pair, et “(2n - 1)/(n+1)” si n est impair</b> +</p> + + +<p> +<b>On définit un ensemble F et G : F = { (2n + 1)/ (n+1) , n ∈ 2k}, G = { (2n - 1)/(n+1), n ∈ 2k+1}</b> +</p> + + +<p> +<b>Donc E = F ∪ G</b> +</p> +</div> +<ul class="org-ul"> +<li><a id="org600d86b"></a>Borne inférieure :<br /> +<div class="outline-text-5" id="text-org600d86b"> +<p> +Inf(E) = Min(inf(F), inf(G)) +</p> + + +<p> +Inf(F) = 1 ; Inf(G) = -1 +</p> + + +<p> +<b>Inf(E)= -1</b> +</p> +</div> +</li> +<li><a id="orgba6c1f0"></a>Borne supérieure :<br /> +<div class="outline-text-5" id="text-orgba6c1f0"> +<p> +Sup(E) = Max(sup(F), sup(G)) +</p> + + +<p> +sup(F) = +∞ ; sup(G) = +∞ +</p> + + +<p> +<b>Sup(E)= +∞</b> +</p> +</div> +</li> +<li><a id="orgc3c7881"></a>Minimum :<br /> +<div class="outline-text-5" id="text-orgc3c7881"> +<p> +Min(E)= -1 +</p> +</div> +</li> +<li><a id="org8d1ee35"></a>Maximum :<br /> +<div class="outline-text-5" id="text-org8d1ee35"> +<p> +E n’as pas de maximum +</p> +</div> +</li> +</ul> +</div> +</div> +<div id="outline-container-org738c8ed" class="outline-3"> +<h3 id="org738c8ed">Exo 2 :</h3> +<div class="outline-text-3" id="text-org738c8ed"> +</div> +<div id="outline-container-org3384f69" class="outline-4"> +<h4 id="org3384f69">Ensemble A :</h4> +<div class="outline-text-4" id="text-org3384f69"> +<p> +A = {x ∈ ℝ , 0 < x <√3} +</p> +</div> +<ul class="org-ul"> +<li><a id="orge3c7b72"></a>Borné<br /> +<div class="outline-text-5" id="text-orge3c7b72"> +<p> +<b>Oui</b>, Inf(A)= 0 ; Sup(A)=√3 +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-orgbbecb74" class="outline-4"> +<h4 id="orgbbecb74">Ensemble B :</h4> +<div class="outline-text-4" id="text-orgbbecb74"> +<p> +B = { x ∈ ℝ , 1/2 < sin x <√3/2} ; +</p> +</div> +<ul class="org-ul"> +<li><a id="org563c4c7"></a>Borné<br /> +<div class="outline-text-5" id="text-org563c4c7"> +<p> +<b>∀ x ∈ B, sin x > 1/2 ∴ Inf(B)= 1/2</b> +</p> + + +<p> +<b>∀ x ∈ B, sin x < √3/2 ∴ Sup(B)= √3/2</b> +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-org9718c4a" class="outline-4"> +<h4 id="org9718c4a">Ensemble C :</h4> +<div class="outline-text-4" id="text-org9718c4a"> +<p> +C = {x ∈ ℝ , x³ > 3} +</p> +</div> +<ul class="org-ul"> +<li><a id="org7b5f08b"></a>Minoré<br /> +<div class="outline-text-5" id="text-org7b5f08b"> +<p> +<b>∀ x ∈ C, x³ > 3 ∴ Inf(C)= 3</b> +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-org43e1433" class="outline-4"> +<h4 id="org43e1433">Ensemble D :</h4> +<div class="outline-text-4" id="text-org43e1433"> +<p> +D = {x ∈ ℝ , e^x < 1/2} +</p> +</div> +<ul class="org-ul"> +<li><a id="orga33ed32"></a>Borné<br /> +<div class="outline-text-5" id="text-orga33ed32"> +<p> +<b>∀ x ∈ C, e^x > 0 ∴ Inf(C)= 0</b> +</p> + + +<p> +<b>∀ x ∈ C, e^x < 1/2 ∴ Sup(C)= 1/2</b> +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-org968305a" class="outline-4"> +<h4 id="org968305a">Ensemble E :</h4> +<div class="outline-text-4" id="text-org968305a"> +<p> +E = {x ∈ ℝ , ∃ p ∈ ℕ* : x = √2/p} +</p> +</div> +<ul class="org-ul"> +<li><a id="org43c7a7b"></a>Majoré<br /> +<div class="outline-text-5" id="text-org43c7a7b"> +<p> +p = √2/x . Donc : <b>Sup(E)=1</b> +</p> +</div> +</li> +</ul> +</div> +</div> +<div id="outline-container-org034d6b3" class="outline-3"> +<h3 id="org034d6b3">Exo 3 :</h3> +<div class="outline-text-3" id="text-org034d6b3"> +<p> +U0 = 3/2 ; U(n+1) = (Un - 1)² + 1 +</p> +</div> +<div id="outline-container-org0e55fe7" class="outline-4"> +<h4 id="org0e55fe7">Question 1 :</h4> +<div class="outline-text-4" id="text-org0e55fe7"> +<p> +Montrer que : ∀ n ∈ ℕ , 1 < Un < 2 . +</p> + + +<p> +<b>(Un - 1)² ≥ 0 <i>Parce que c’est un carré</i></b> +</p> + + +<p> +<b>(Un - 1)² + 1 > 1</b> ; <b>U(n+1) ≥ 1</b> +</p> +</div> +<ul class="org-ul"> +<li><a id="org354cb33"></a>Raisonnement par récurrence :<br /> +<div class="outline-text-5" id="text-org354cb33"> +<p> +P(n) : ∀ n ∈ ℕ ; 1 < Un < 2 +</p> + + +<p> +P(0) est vraie : 1 < 3/2 < 2 +</p> + + +<p> +On suppose que P(n) est vraie et on vérifie P(n+1) pour une contradiction +</p> + + +<p> +1< Un < 2 ; 0 < Un - 1 < 1 ; 0 < (Un - 1)² < 1 ; 1 < (Un - 1)² + 1< 2 ; <b>1 < U(n+1) < 2</b> Donc elle est correcte +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-orgce11f0b" class="outline-4"> +<h4 id="orgce11f0b">Question 2 :</h4> +<div class="outline-text-4" id="text-orgce11f0b"> +<p> +Montrer que (Un)n est strictement monotone : +</p> + + +<p> +<b>U(n+1) - Un = (Un - 1)² + 1 - Un</b> ; <b>U(n+1) - Un = Un² + 1 - 2Un + 1 - Un</b> ; <b>U(n+1) - Un = Un² - 3Un + 2</b> +</p> + + +<p> +On étudie <b>Un² - 3Un + 2</b> sur l’intervalle ]1, 2[ : Un² - 3Un + 2 = 0 est une équation du 2nd ordre, <b>Δ = 1</b> , elle accepte deux solutions : Un = 1 et Un = 2 +</p> + + +<p> +On déduit que <b>Un² - 3Un + 2</b> est négatif sur [1 , 2] et positif en dehors, donc <b>∀ 1 < Un < 2 , Un² - 3Un + 2 < 0</b> ; <b>∀ 1 < Un < 2 , U(n+1) - Un < 0</b> ; <b>∀ 1 < Un < 2 , U(n+1) < Un</b> Donc (Un)n est une suite strictement monotonne décroissante +</p> +</div> +</div> +</div> +</div> +<div id="outline-container-org0b95370" class="outline-2"> +<h2 id="org0b95370">4th cours (Suite) : <i>Oct 10</i></h2> +<div class="outline-text-2" id="text-org0b95370"> +</div> +<div id="outline-container-org37a71e2" class="outline-3"> +<h3 id="org37a71e2">Les suites convergentes</h3> +<div class="outline-text-3" id="text-org37a71e2"> +<p> +Soit (Un)n est une suite convergente si lim Un n–> +∞ = l +</p> +</div> +<div id="outline-container-org4a6d045" class="outline-4"> +<h4 id="org4a6d045">Remarque :</h4> +<div class="outline-text-4" id="text-org4a6d045"> +<ol class="org-ol"> +<li>Un est une suite convergente alors Un est bornee</li> +<li>Un est une suite convergente lim Un n—> +∞ = l ⇔ lim |Un| n—> +∞ = |l|</li> +<li>Un est une suite majoree et croissante ⇒ Un converge</li> +<li>Un est une suite minoree et decroissante ⇒ Un converge</li> +<li>Soient (Un) et (Vn) deux suites convergentes, alors +<ol class="org-ol"> +<li>Un + Vn est convergente</li> +<li>Un * Vn est convergente</li> +<li>∀λ ∈ ℝ , (λUn) converge</li> +</ol></li> +<li>Soit Un est une suite bornee et soit Vn une suite. lim Vn n->+∞ = 0 Alors lim Vn * Un n-> +∞ = 0</li> +</ol> +</div> +</div> +</div> +<div id="outline-container-orgcd9ef0a" class="outline-3"> +<h3 id="orgcd9ef0a">Theoreme d’encadrement</h3> +<div class="outline-text-3" id="text-orgcd9ef0a"> +<p> +Soient Un Vn et Wn trois suites ∀n ∈ ℕ, Un ≤ Vn ≤ Wn . et lim Un n->∞ = lim Wn n-> +∞ = l ⇒ lim Vn n-> +∞ = l +</p> +</div> +</div> +</div> +</div> +<div id="postamble" class="status"> +<p class="author">Author: Crystal</p> +<p class="date">Created: 2023-10-11 Wed 19:18</p> +</div> +</body> +</html> \ No newline at end of file diff --git a/uni_notes/architecture.html b/uni_notes/architecture.html new file mode 100755 index 0000000..b5ba235 --- /dev/null +++ b/uni_notes/architecture.html @@ -0,0 +1,899 @@ +<?xml version="1.0" encoding="utf-8"?> +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" +"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> +<head> +<!-- 2023-10-11 Wed 19:53 --> +<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> +<meta name="viewport" content="width=device-width, initial-scale=1" /> +<title>Architecture 1</title> +<meta name="author" content="Crystal" /> +<meta name="generator" content="Org Mode" /> +<link rel="stylesheet" type="text/css" href="../src/css/colors.css"/> +<link rel="stylesheet" type="text/css" href="../src/css/style.css"/> +</head> +<body> +<div id="content" class="content"> +<h1 class="title">Architecture 1</h1> +<div id="outline-container-org644596f" class="outline-2"> +<h2 id="org644596f">Premier cours : Les systémes de numération <i>Sep 27</i> :</h2> +<div class="outline-text-2" id="text-org644596f"> +<p> +Un système de numération est une méthode pour représenter des nombres à l’aide de symboles et de règles. Chaque système, comme le décimal (base 10) ou le binaire (base 2), utilise une base définie pour représenter des valeurs numériques. Il est caractérisé par 3 entitiés mathématiques importantes: +</p> + +<ol class="org-ol"> +<li>Une base (genre 10, ou 2)</li> +<li>Un ensemble de chiffres</li> +<li>Des régles de représentations des nombres</li> +</ol> +</div> +<div id="outline-container-org93dcedb" class="outline-4"> +<h4 id="org93dcedb"><b>Examples :</b></h4> +<div class="outline-text-4" id="text-org93dcedb"> +<p> +<i>B10 est un systéme de numération caractérisé par:</i> +</p> +<ul class="org-ul"> +<li>Base = 10</li> +<li>Un ensemble de chiffres : (0,1,2,3,4,5,6,7,8,9)</li> +</ul> + +<p> +<i>B16 est un autre systéme de numération caractérisé par:</i> +</p> +<ul class="org-ul"> +<li>Base = 16</li> +<li><p> +Un ensemble de chiffres : (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F) +</p> + +<p> +<b>Puisse-qu’on peut pas utiliser des nombres a deux chiffres, on utilise des lettres aprés 9, en leur donnant des valeurs tel que :</b> +</p> + +<p> +A : 10 ; B : 11 ; C : 12 ; D : 13 ; E : 14 ; F : 15 +</p></li> +</ul> +</div> +</div> +<div id="outline-container-orgc27c000" class="outline-3"> +<h3 id="orgc27c000">Comment passer d’un systéme a base 10 a un autre</h3> +<div class="outline-text-3" id="text-orgc27c000"> +<p> +On symbolise un chiffre dans la base x par : (Nombre)x +</p> +</div> +<div id="outline-container-org54f4213" class="outline-4"> +<h4 id="org54f4213">Pour les chiffres entiers :</h4> +<div class="outline-text-4" id="text-org54f4213"> +<p> +<b>On fait une division successive, on prends le nombre 3257 comme exemple, on veut le faire passer d’une base décimale á une base 16:</b> +</p> + + +<p> +(3257)10 -—> (?)16 +</p> + + +<p> +On dévise 3257 par 16, et les restants de la division serra la valeur en base16: +</p> + +<p> +3257/16 = 203 + <b>9</b> / 16 +</p> + +<p> +203/16 = 12 + <b>B</b> / 16 <i>REMARQUE, 11 N’APPARTIENS PAS A L’ENSEMBLE DES CHIFFRES EN BASE16, CE QUI VEUT DIRE QU’ON LE REMPLACE PAR SON EQUIVALENT, DANS CE CAS LA: <b>B</b></i> +</p> + +<p> +12/16 = 0 + <b>C</b> / 16 <i>Pareil ici, 12 n’existe pas, donc c’est C. Autre note : La division s’arréte quand le résultat de la division est nul</i> +</p> +</div> +<ul class="org-ul"> +<li><a id="orgc41ba5c"></a><b>Conclusion:</b><br /> +<div class="outline-text-5" id="text-orgc41ba5c"> +<p> +(3257)10 -—> (CB9)16 +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-org6f8ff12" class="outline-4"> +<h4 id="org6f8ff12">Pour les chiffres non entiers :</h4> +<div class="outline-text-4" id="text-org6f8ff12"> +<p> +<b>On fait la division successive pour la partie entiére, et une multiplication successive pour la partie rationelle:</b> +</p> + +<p> +(3257,32)10 -—> (?)16 +</p> + +<p> +On a déja la partie entiére donc on s’occupe de la partie aprés la virgule: +</p> + +<p> +0,32 x 16 = <b>5</b>,12 +</p> + +<p> +0,12 x 16 = <b>1</b>,92 +</p> + +<p> +0,92 x 16 = <b>E</b>,72 <i>On a pas de 15 donc c’est un E</i> +</p> + +<p> +0,72 x 16 = <b>B</b>,52 +</p> + +<p> +0,52 x 16 = <b>8</b>,32 +</p> + +<p> +0,32 x 16 = <b>5</b>,12 +</p> + +<p> +… +</p> + +<p> +<i>On s’arréte quand on trouve un chiffre entier, et si on trouve pas, on s’arréte quand on remarque une répetition, dans ce cas la, la séquance 51EB8 vas se répéter indéfiniment, donc on se contente d’écrire la partie qui se répéte avec une barre en haut</i> +</p> + + +<p> +(3257,32)10 -—> (CB9, <span class="underline">51EB8</span>)16 +</p> +</div> +</div> +</div> +</div> +<div id="outline-container-orga79d939" class="outline-2"> +<h2 id="orga79d939">2nd cours : Les systèmes de numération (Suite) <i>Oct 3</i> :</h2> +<div class="outline-text-2" id="text-orga79d939"> +</div> +<div id="outline-container-org3e165a8" class="outline-3"> +<h3 id="org3e165a8">Comment passer d’une base N a la base 10 :</h3> +<div class="outline-text-3" id="text-org3e165a8"> +<p> +Prenons comme exemple le nombre (11210,0011)3 , chaque chiffre dans ce nombre a un rang qui commence par 0 au premier chiffre (a gauche de la virgule) et qui augmente d’un plus qu’on avance a gauche, et diminue si on part a droite. Dans ce cas la : +</p> + + +<p> +(11210,0011)3 ; le 0 est de rang 0, le 1 est de rang 1, le 2 est de rang 2, le 1 est de rang 3, le 1 est de rang 4. Et si on part du coté de la virgule, 0 est de rang -1, 0 est de rang -2, le 1 est de rang -3, et le 1 est de rang -4. +</p> + + +<p> +Et pour passer a la base 10, il suffit d’appliquer cette formule : <b>Chiffre x Base^(rang) + 2emeChiffre x Base^(rang)… etc</b>, donc dans notre example: +</p> + + +<p> +<i>0 x 3° + 1 x 3¹ + 2 x 3² + 1 x 3³ + 1 x 3^4 + 0 x 3¯¹ + 0 x 3¯² + 1 x 3¯³ + 1 x 3^(-4) ≈ (129,05)10</i> +</p> +</div> +</div> +<div id="outline-container-org5a3bcfe" class="outline-3"> +<h3 id="org5a3bcfe">Comment passer d’une base N a une base N^(n) :</h3> +<div class="outline-text-3" id="text-org5a3bcfe"> +<p> +Si il ya une relation entre une base et une autre, on peut directement transformer vers cette base. +</p> +</div> +<div id="outline-container-orgab808d0" class="outline-4"> +<h4 id="orgab808d0">Exemple :</h4> +<div class="outline-text-4" id="text-orgab808d0"> +<p> +Pour passer de la base 2 a la base 8 (8 qui est 2³) on découpe les chiffres 3 par 3 +</p> + + +<p> +(1 101 011, 011)2 ; Pour le dernier 1 qui est seul <code>tout comme moi</code> il suffit d’ajouter des 0 à gauche (car on peut) pour compléter le découpage. +</p> + + +<p> +(001 101 011, 011)2; Next step c’est de dessiner le tableau de conversion de la base 2 a la base 8 ( un tableau a 3 bits ) +</p> + + +<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> + + +<colgroup> +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> +</colgroup> +<thead> +<tr> +<th scope="col" class="org-right">N</th> +<th scope="col" class="org-right"> </th> +<th scope="col" class="org-right"> </th> +<th scope="col" class="org-right"> </th> +</tr> +</thead> +<tbody> +<tr> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +</tr> + +<tr> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +</tr> + +<tr> +<td class="org-right">2</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +</tr> + +<tr> +<td class="org-right">3</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +</tr> + +<tr> +<td class="org-right">4</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +</tr> + +<tr> +<td class="org-right">5</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +</tr> + +<tr> +<td class="org-right">6</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +</tr> + +<tr> +<td class="org-right">7</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +</tr> +</tbody> +</table> + + +<p> +Pour remplir on a qu’a diviser les chiffres en deux, et mettre des 0 dans la première partie et des 1 dans la 2éme, et en faire de même pour les autres colonnes . +</p> + + +<p> +Maintenant il suffit de trouver l’équivalent de la base2 en base8 : +</p> + + +<p> +001 c’est 1 ; 101 c’est 5 ; 011 c’est 3 ; donc <b>(1101011,011)2 —> (153,3)8</b> +</p> +</div> +</div> +</div> +<div id="outline-container-orgbf931c5" class="outline-3"> +<h3 id="orgbf931c5">L’arithmétique binaire :</h3> +<div class="outline-text-3" id="text-orgbf931c5"> +</div> +<div id="outline-container-orgbe5bb2f" class="outline-4"> +<h4 id="orgbe5bb2f">L’addition :</h4> +<div class="outline-text-4" id="text-orgbe5bb2f"> +<p> +0 + 0 = 0 On retiens 0 +</p> + + +<p> +1 + 0 = 1 On retiens 0 +</p> + + +<p> +0 + 1 = 1 On retiens 0 +</p> + + +<p> +1 + 1 = 0 On retiens 1 +</p> + + +<p> +1 + 1 + 1 = 1 On retiens 1 +</p> + + +<p> +Donc 0110 + 1101 = 10011 +</p> +</div> +</div> +<div id="outline-container-org76fafea" class="outline-4"> +<h4 id="org76fafea">La soustraction :</h4> +<div class="outline-text-4" id="text-org76fafea"> +<p> +0 - 0 = 0 On emprunt = 0 +</p> + + +<p> +1 - 0 = 1 On emprunt = 0 +</p> + + +<p> +0 - 1 = 1 On emprunt = 1 +</p> + + +<p> +1 - 1 = 0 On emprunt = 0 +</p> +</div> +</div> +</div> +<div id="outline-container-org0db7791" class="outline-3"> +<h3 id="org0db7791">TP N°1 :</h3> +<div class="outline-text-3" id="text-org0db7791"> +</div> +<div id="outline-container-org602a6f4" class="outline-4"> +<h4 id="org602a6f4">Exo1:</h4> +<div class="outline-text-4" id="text-org602a6f4"> +<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> + + +<colgroup> +<col class="org-left" /> + +<col class="org-left" /> + +<col class="org-left" /> + +<col class="org-left" /> + +<col class="org-left" /> +</colgroup> +<thead> +<tr> +<th scope="col" class="org-left">Base 10</th> +<th scope="col" class="org-left">Base 2</th> +<th scope="col" class="org-left">Base 3</th> +<th scope="col" class="org-left">Base 8</th> +<th scope="col" class="org-left">Base 16</th> +</tr> +</thead> +<tbody> +<tr> +<td class="org-left">22,75</td> +<td class="org-left">10110,11</td> +<td class="org-left">211, <span class="underline">20</span></td> +<td class="org-left">26,6</td> +<td class="org-left">F6,C</td> +</tr> + +<tr> +<td class="org-left">684,125</td> +<td class="org-left">1010101100,001</td> +<td class="org-left">221100, <span class="underline">01</span></td> +<td class="org-left">1254,1</td> +<td class="org-left">2AC,2</td> +</tr> + +<tr> +<td class="org-left">3931,625</td> +<td class="org-left">111101011011,101</td> +<td class="org-left">1101121, <span class="underline">12</span></td> +<td class="org-left">7533,5</td> +<td class="org-left">F5B,A</td> +</tr> + +<tr> +<td class="org-left">52,38</td> +<td class="org-left">110100,011</td> +<td class="org-left">1221,101</td> +<td class="org-left">64,3</td> +<td class="org-left">34,6147</td> +</tr> + +<tr> +<td class="org-left">10,67</td> +<td class="org-left">1010,101</td> +<td class="org-left">23,5</td> +<td class="org-left">12,5</td> +<td class="org-left">A,AB85</td> +</tr> +</tbody> +</table> +</div> +<ul class="org-ul"> +<li><a id="org44fc05c"></a>(10110,11)2<br /> +<div class="outline-text-5" id="text-org44fc05c"> +<p> +0 x 2° + 1 x 2¹ + 1 x 2² + 0 x 2³ + 1 x 2^(4) + 1 x 2¯¹ + 1 x 2¯² = (22.75)10 +</p> +</div> +<ul class="org-ul"> +<li><a id="org6e587a2"></a>(22,75)10 -—> (3)<br /> +<div class="outline-text-6" id="text-org6e587a2"> +<p> +22/3 = 7 R <b>1</b> ; 7/3 = 2 R <b>1</b> ; 2/3 = 0 R <b>2</b> +</p> + + +<p> +0,75 x 3 = <b>2</b>.25 ; 0,25 x 3 = <b>0</b>.75 ….. +</p> + + +<p> +(22,75)10 -—> (211, <span class="underline">20</span>) +</p> +</div> +</li> +<li><a id="org5ed87ae"></a>(10110,11)2 -—> (8)<br /> +<div class="outline-text-6" id="text-org5ed87ae"> +<p> +8 = 2³ ; (010 110,110)2 -—> (?)8 +</p> + + +<p> +En utilisant le tableau 3bits : +</p> + + +<p> +010 : 2 ; 110 : 6 ; 110 : 6 +</p> + + +<p> +(10110,11)2 -—> (26,6)8 +</p> +</div> +</li> +<li><a id="org941eaad"></a>(22,75)10 -—> (16)<br /> +<div class="outline-text-6" id="text-org941eaad"> +<p> +22/16 = 1 R <b>6</b> ; 1/16 : 0 R <b>F</b> +</p> + + +<p> +0,75 x 16 = <b>C</b> +</p> + + +<p> +(22,75)10 -—> (F6,C)16 +</p> +</div> +</li> +</ul> +</li> +<li><a id="org14d08d6"></a>(1254,1)8<br /> +<div class="outline-text-5" id="text-org14d08d6"> +<p> +4 x 8° + 5 x 8¹ + 2 x 8² + 1 x 8³ + 1 x 8¯¹ = (684,125)10 +</p> +</div> +<ul class="org-ul"> +<li><a id="orgd83dbd8"></a>(1254,1)8 -—> (?)2<br /> +<div class="outline-text-6" id="text-orgd83dbd8"> +<p> +En utilisant le tableau 3bits : +</p> + + +<p> +001 010 101 100,001 <i>We will get rid of the leading zeros</i> +</p> + + +<p> +(1010101100,001)2 +</p> +</div> +</li> +<li><a id="org1897d7c"></a>(684,125)10 -—> (?)3<br /> +<div class="outline-text-6" id="text-org1897d7c"> +<p> +684/3 = 228 R <b>0</b> ; 228/3 = 76 R <b>0</b> ; 76/3 = 25 R <b>1</b> ; 25/3 = 8 R <b>1</b> ; 8/3 = 2 R <b>2</b> ; 2/3 = 0 R <b>2</b> +</p> + + +<p> +0,125 x 3 = <b>0</b>,375 ; 0,375 x 3 = <b>1</b>,125 +</p> + + +<p> +(221100, <span class="underline">01</span>)3 +</p> +</div> +</li> +<li><a id="org5d6fc0d"></a>(684,125)10 -—> (?)16<br /> +<div class="outline-text-6" id="text-org5d6fc0d"> +<p> +684/16 = 42 R <b>C</b> ; 42/16 = 2 R <b>A</b> ; 2/16 0 R <b>2</b> +</p> + + +<p> +0,125 x 16 = <b>2</b> +</p> + + +<p> +(2AC,2)16 +</p> +</div> +</li> +</ul> +</li> +<li><a id="orgdb748f1"></a>(F5B,A)16<br /> +<div class="outline-text-5" id="text-orgdb748f1"> +<p> +11 x 16° + 5 x 16 + 15 x 16² + 10 x 16¯¹ = (3931,625)10 +</p> +</div> +<ul class="org-ul"> +<li><a id="org68f19f3"></a>(3931,625)10 -—> (8)<br /> +<div class="outline-text-6" id="text-org68f19f3"> +<p> +3931/8 = 491 R <b>3</b> ; 491/8 = 61 R <b>3</b> ; 61/8 = 7 R <b>5</b> ; 7/8 = 0 R <b>7</b> +</p> + + +<p> +0,625 x 8 = <b>5</b> +</p> + + +<p> +(7533,5)8 +</p> +</div> +</li> +<li><a id="org39a13c6"></a>(7533,5)8 -—> (2)<br /> +<div class="outline-text-6" id="text-org39a13c6"> +<p> +En utilisant le tableau 3bits +</p> + +<p> +(111 101 011 011,101)2 +</p> +</div> +</li> +<li><a id="org8613276"></a>(3931,625)10 -—> (3)<br /> +<div class="outline-text-6" id="text-org8613276"> +<p> +3931/3 = 1310 R <b>1</b> ; 1310/3 = 436 R <b>2</b> ; 436/3 = 145 R <b>1</b> ; 145/3 = 48 R <b>1</b> ; 48/3 = 16 R <b>0</b> ; 16/3 = 5 R <b>1</b> ; 5/3 = 1 R <b>2</b> ; 1/3 = 0 R <b>1</b> +</p> + + +<p> +0.625 x 3 = <b>1</b>,875 ; 0,875 x 3 = <b>2</b>,625 +</p> + + +<p> +(1101121, <span class="underline">12</span>)3 +</p> +</div> +</li> +</ul> +</li> +<li><a id="orgf5ad2f0"></a>(52,38)10<br /> +<div class="outline-text-5" id="text-orgf5ad2f0"> +<p> +52/2 = 26 R <b>0</b> ; 26/2 = 13 R <b>0</b> ; 13/2 = 6 R <b>1</b> ; 6/2 = 3 R <b>0</b> ; 3/2 = 1 R <b>1</b> ; 1/2 = 0 R <b>1</b> +</p> + + +<p> +0,38 x 2 = <b>0</b>,76 ; 0,76 x 2 = <b>1</b>,52 ; 0,52 x 2 = <b>1</b>,04 ; 0,04 x 2 = <b>0</b>,08 …. +</p> + + +<p> +(110100,0110)2 +</p> +</div> +<ul class="org-ul"> +<li><a id="org40d1958"></a>(52,38)10 -—> (3)<br /> +<div class="outline-text-6" id="text-org40d1958"> +<p> +52/3 = 17 R <b>1</b> ; 17/3 = 5 R <b>2</b> ; 5/3 = 1 R <b>2</b> ; 1/3 = 0 R <b>1</b> +</p> + + +<p> +0,38 x 3 = <b>1</b>.14 ; 0,14 x 3 = <b>0</b>.42 ; 0,42 x 3 = <b>1</b>.26 ; 0.26 x 3 = <b>0</b>.78 … +</p> + + +<p> +(1221,101)3 +</p> +</div> +</li> +<li><a id="org15b5aba"></a>(110100,011)2 -—> (8)<br /> +<div class="outline-text-6" id="text-org15b5aba"> +<p> +En utilisant le tableau 3bits: +</p> + + +<p> +(110 100,011)2 -—> (64,3)8 +</p> +</div> +</li> +<li><a id="org495819e"></a>(52,38)10 -—> (16)<br /> +<div class="outline-text-6" id="text-org495819e"> +<p> +52/16 = 3 R <b>4</b> ; 3/16 = 0 R <b>3</b> +</p> + + +<p> +0,38 x 16 = <b>6</b>,08 ; 0,08 x 16 = <b>1</b>,28 ; 0,28 x 16 = <b>4</b>,48 ; 0,48 x 16 = <b>7</b>,68 …. +</p> + + +<p> +(34,6147)16 +</p> +</div> +</li> +</ul> +</li> +<li><a id="orga49bba8"></a>(23,5)3<br /> +<div class="outline-text-5" id="text-orga49bba8"> +<p> +3 x 3° + 2 x 3 + 5 x 3¯¹ = (10.67)10 +</p> +</div> +<ul class="org-ul"> +<li><a id="org2b7d27e"></a>(10,67)10 -—> (2)<br /> +<div class="outline-text-6" id="text-org2b7d27e"> +<p> +10/2 = 5 R <b>0</b> ; 5/2 = 2 R <b>1</b> ; 2/2 = 1 R <b>0</b> ; 1/2 = 0 R <b>1</b> +</p> + + +<p> +0,67 x 2 = <b>1</b>,34 ; 0,34 x 2 = <b>0</b>,68 ; 0,68 x 2 = <b>1</b>,36 ; 0,36 x 2 = <b>0</b>,72 … +</p> + + +<p> +(1010,101)2 +</p> +</div> +</li> +<li><a id="org7aad862"></a>(001 010,101)2 -—> (8)<br /> +<div class="outline-text-6" id="text-org7aad862"> +<p> +<b>Ô Magic 3bits table, save me soul, me children and me maiden:</b> +</p> + + +<p> +(12,5)8 +</p> +</div> +</li> +<li><a id="orgdb90500"></a>(10,67)10 -—> (16)<br /> +<div class="outline-text-6" id="text-orgdb90500"> +<p> +10/16 = 0 R <b>A</b> +</p> + + +<p> +0,67 x 16 = <b>A</b>,72 ; 0,72 x 16 = <b>B</b>,52 ; 0,52 x 16 = <b>8</b>,32 ; 0,32 x 16 = <b>5</b>,12 … +</p> + + +<p> +(A,AB85)16 +</p> +</div> +</li> +</ul> +</li> +</ul> +</div> +<div id="outline-container-orgd9aad1c" class="outline-4"> +<h4 id="orgd9aad1c">Exo2:</h4> +<div class="outline-text-4" id="text-orgd9aad1c"> +</div> +<ul class="org-ul"> +<li><a id="org9a44306"></a>(34)? = (22)10<br /> +<div class="outline-text-5" id="text-org9a44306"> +<p> +(34)a = (22)10 ; 4 x a° + 3 x a = 22 ; 4 + 3a = 22 ; 3a = 18 +</p> + + +<p> +<b>a = 6</b> +</p> +</div> +</li> +<li><a id="org621cd70"></a>(75)? = (117)10<br /> +<div class="outline-text-5" id="text-org621cd70"> +<p> +(75)b = (117)10 ; 5 x b° + 7 x b¹ = 117 ; 5 + 7b = 117 ; 7b = 112 +</p> + + +<p> +<b>b = 16</b> +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-org25943b2" class="outline-4"> +<h4 id="org25943b2">Exo3:</h4> +<div class="outline-text-4" id="text-org25943b2"> +</div> +<ul class="org-ul"> +<li><a id="orge98e1b3"></a>(101011)2 + (111011)2<br /> +<div class="outline-text-5" id="text-orge98e1b3"> +<p> +101011 + 111011 = 1100110 +</p> +</div> +</li> +<li><a id="org9336956"></a>(1011,1101)2 + (11,1)2<br /> +<div class="outline-text-5" id="text-org9336956"> +<p> +1011,1101 + 11,1000 = 1111,0101 +</p> +</div> +</li> +<li><a id="org8a605df"></a>(1010,0101)2 - (110,1001)2<br /> +<div class="outline-text-5" id="text-org8a605df"> +<p> +1010,0101 - 110,1001 = 11,1100 +</p> +</div> +</li> +</ul> +</div> +</div> +<div id="outline-container-orgd1bb996" class="outline-3"> +<h3 id="orgd1bb996">L’arithmétique binaire (Suite): <i>Oct 4</i></h3> +<div class="outline-text-3" id="text-orgd1bb996"> +</div> +<div id="outline-container-org1a9b0f2" class="outline-4"> +<h4 id="org1a9b0f2">La multiplication :</h4> +<div class="outline-text-4" id="text-org1a9b0f2"> +<p> +0 x 0 = 0 +</p> + + +<p> +0 x 1 = 0 +</p> + + +<p> +1 x 0 = 0 +</p> + + +<p> +1 x 1 = 1 +</p> +</div> +</div> +<div id="outline-container-org17f1eab" class="outline-4"> +<h4 id="org17f1eab">La division :</h4> +<div class="outline-text-4" id="text-org17f1eab"> +<p> +On divise de la manière la plus normale du monde !!! +</p> +</div> +</div> +</div> +</div> +<div id="outline-container-org4e457be" class="outline-2"> +<h2 id="org4e457be">4th cours : Le codage <i>Oct 10</i></h2> +<div class="outline-text-2" id="text-org4e457be"> +</div> +<div id="outline-container-org8c7cda2" class="outline-3"> +<h3 id="org8c7cda2">Le codage des entiers positifs</h3> +<div class="outline-text-3" id="text-org8c7cda2"> +<p> +Le codage sur n bits permet de representer tout les entiers naturels compris entre [0, 2^n - 1]. On peut coder sur 8bits les entiers entre [0;2^8 - 1(255)] +</p> +</div> +</div> +<div id="outline-container-org1aa2cfc" class="outline-3"> +<h3 id="org1aa2cfc">Le codage des nombres relatifs</h3> +<div class="outline-text-3" id="text-org1aa2cfc"> +</div> +<div id="outline-container-orgc29141b" class="outline-4"> +<h4 id="orgc29141b">Remarque</h4> +<div class="outline-text-4" id="text-orgc29141b"> +<p> +Quelque soit le codage utilise, par convention le dernier bit est reserve pour le signe. ou 1 est negatif et 0 est positif. +</p> +</div> +</div> +<div id="outline-container-org4899ed4" class="outline-4"> +<h4 id="org4899ed4">Le codage en signe + valeur absolue (SVA):</h4> +<div class="outline-text-4" id="text-org4899ed4"> +<p> +Avec n bits le n eme est reserve au signe : [-(2^n-1)-1 , 2^n-1 -1]. Sur 8bits [-127, 127] +</p> +</div> +</div> +<div id="outline-container-orgc410845" class="outline-4"> +<h4 id="orgc410845">Codage en compliment a 1 (CR):</h4> +<div class="outline-text-4" id="text-orgc410845"> +<p> +On obtiens le compliment a 1 d’un nombre binaire en inversant chaqu’un de ses bits (1 -> 0 et 0-> 1) les nombres positifs sont la meme que SVA (il reste tel qu’il est) +</p> +</div> +</div> +<div id="outline-container-orgb19633c" class="outline-4"> +<h4 id="orgb19633c">Codage en compliment a 2 (CV):</h4> +<div class="outline-text-4" id="text-orgb19633c"> +<p> +C’est literallement CR + 1 pour les negatifs et SVA pour les nombres positifs +</p> +</div> +</div> +</div> +</div> +</div> +<div id="postamble" class="status"> +<p class="author">Author: Crystal</p> +<p class="date">Created: 2023-10-11 Wed 19:53</p> +</div> +</body> +</html> \ No newline at end of file |
