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diff --git a/articles/discord.html b/articles/discord.html index 1d93966..4eff0d0 100755 --- a/articles/discord.html +++ b/articles/discord.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <head> -<!-- 2023-11-01 Wed 20:09 --> +<!-- 2023-11-01 Wed 20:16 --> <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> @@ -11,14 +11,14 @@ <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"/> -<link rel="icon" type="image/x-icon" href="../../favicon.ico"> +<link rel="icon" type="image/x-icon" href="../../favicon.png"> </head> <body> <div id="content" class="content"> <h1 class="title"><b><b>Discord</b></b> : an internet cancer</h1> -<div id="outline-container-org798d85b" class="outline-2"> -<h2 id="org798d85b">Preamble</h2> -<div class="outline-text-2" id="text-org798d85b"> +<div id="outline-container-orgfef773e" class="outline-2"> +<h2 id="orgfef773e">Preamble</h2> +<div class="outline-text-2" id="text-orgfef773e"> <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> @@ -28,9 +28,9 @@ I also wanted to make it clear that this is PURELY from my personal experience w </p> </div> </div> -<div id="outline-container-orgfcb8000" class="outline-2"> -<h2 id="orgfcb8000">Chapter one : Curiosity</h2> -<div class="outline-text-2" id="text-orgfcb8000"> +<div id="outline-container-orgfc01505" class="outline-2"> +<h2 id="orgfc01505">Chapter one : Curiosity</h2> +<div class="outline-text-2" id="text-orgfc01505"> <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> @@ -52,9 +52,9 @@ So you join that group they invited you to, it could be anything from a small fr </p> </div> </div> -<div id="outline-container-org219978e" class="outline-2"> -<h2 id="org219978e">Chapter two : The hierarchy</h2> -<div class="outline-text-2" id="text-org219978e"> +<div id="outline-container-orgda2040e" class="outline-2"> +<h2 id="orgda2040e">Chapter two : The hierarchy</h2> +<div class="outline-text-2" id="text-orgda2040e"> <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> @@ -109,9 +109,9 @@ If you are paying attention, you would know that all of these will always end wi </p> </div> </div> -<div id="outline-container-orgf7a70c1" class="outline-2"> -<h2 id="orgf7a70c1">Chapter three : <b><b>Discord</b></b> takes a once thriving community and splits it</h2> -<div class="outline-text-2" id="text-orgf7a70c1"> +<div id="outline-container-org0b5656a" class="outline-2"> +<h2 id="org0b5656a">Chapter three : <b><b>Discord</b></b> takes a once thriving community and splits it</h2> +<div class="outline-text-2" id="text-org0b5656a"> <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> @@ -133,9 +133,9 @@ So you basically killed a community, in 6 easy steps !!! And of course this will </p> </div> </div> -<div id="outline-container-orgd8817af" class="outline-2"> -<h2 id="orgd8817af">Chapter four : <b><b>Discord</b></b> users are NOT your friends</h2> -<div class="outline-text-2" id="text-orgd8817af"> +<div id="outline-container-org703841d" class="outline-2"> +<h2 id="org703841d">Chapter four : <b><b>Discord</b></b> users are NOT your friends</h2> +<div class="outline-text-2" id="text-org703841d"> <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? @@ -154,9 +154,9 @@ So, for those pondering the idea of <b><b>Discord</b></b> romance, think twice! </p> </div> </div> -<div id="outline-container-org1e6054d" class="outline-2"> -<h2 id="org1e6054d">Final Chapter : Login-walls</h2> -<div class="outline-text-2" id="text-org1e6054d"> +<div id="outline-container-orgb103108" class="outline-2"> +<h2 id="orgb103108">Final Chapter : Login-walls</h2> +<div class="outline-text-2" id="text-orgb103108"> <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> @@ -182,9 +182,9 @@ In conclusion, <b><b>Discord</b></b>’s penchant for login-walls is like lo </p> </div> </div> -<div id="outline-container-orga97b513" class="outline-2"> -<h2 id="orga97b513">Conclusion</h2> -<div class="outline-text-2" id="text-orga97b513"> +<div id="outline-container-org5df4158" class="outline-2"> +<h2 id="org5df4158">Conclusion</h2> +<div class="outline-text-2" id="text-org5df4158"> <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> @@ -209,7 +209,7 @@ If you ever have more anecdotes, insights, or questions to add to this digital s </div> <div id="postamble" class="status"> <p class="author">Author: Crystal</p> -<p class="date">Created: 2023-11-01 Wed 20:09</p> +<p class="date">Created: 2023-11-01 Wed 20:16</p> </div> </body> </html> \ No newline at end of file diff --git a/favicon.png b/favicon.png new file mode 100644 index 0000000..9aac13b --- /dev/null +++ b/favicon.png Binary files differdiff --git a/index.html b/index.html index 6a9be8a..90063ac 100755 --- a/index.html +++ b/index.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <head> -<!-- 2023-11-01 Wed 20:09 --> +<!-- 2023-11-01 Wed 20:15 --> <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> @@ -11,14 +11,14 @@ <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"/> -<link rel="icon" type="image/x-icon" href="favicon.ico"> +<link rel="icon" type="image/x-icon" href="favicon.png"> </head> <body> <div id="content" class="content"> <h1 class="title">Crystal’s Website 💜</h1> -<div id="outline-container-org91f2c37" class="outline-2"> -<h2 id="org91f2c37">Welcome to the wired</h2> -<div class="outline-text-2" id="text-org91f2c37"> +<div id="outline-container-orgdb39371" class="outline-2"> +<h2 id="orgdb39371">Welcome to the wired</h2> +<div class="outline-text-2" id="text-orgdb39371"> <p> Hi there, <a href="./super_secret.html">adorable you!</a> </p> @@ -29,27 +29,27 @@ And welcome to my little corner of the internet, here I will be posting my rando -<div id="org4cac298" class="figure"> +<div id="org54f919b" class="figure"> <p><img src="./src/gifs/Lain_chibi.png" alt="Lain_chibi.png" width="200px" /> </p> </div> </div> </div> -<div id="outline-container-org2d407df" class="outline-2"> -<h2 id="org2d407df">Articles ( NEW !!!! )</h2> -<div class="outline-text-2" id="text-org2d407df"> +<div id="outline-container-orga75bec3" class="outline-2"> +<h2 id="orga75bec3">Articles ( NEW !!!! )</h2> +<div class="outline-text-2" id="text-orga75bec3"> <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-org3403ba9" class="outline-2"> -<h2 id="org3403ba9">root@localhost $ whoami</h2> -<div class="outline-text-2" id="text-org3403ba9"> +<div id="outline-container-org8f4a35c" class="outline-2"> +<h2 id="org8f4a35c">root@localhost $ whoami</h2> +<div class="outline-text-2" id="text-org8f4a35c"> </div> -<div id="outline-container-orgaeb7592" class="outline-3"> -<h3 id="orgaeb7592">About me :</h3> -<div class="outline-text-3" id="text-orgaeb7592"> +<div id="outline-container-org7a12c9c" class="outline-3"> +<h3 id="org7a12c9c">About me :</h3> +<div class="outline-text-3" id="text-org7a12c9c"> <ul class="org-ul"> <li>Name : <b>Crystal</b></li> <li>Age : <b>18 years old</b></li> @@ -68,9 +68,9 @@ If you want to contact me (which would be really surprising) contact me via <a h </p> </div> </div> -<div id="outline-container-org24573ef" class="outline-3"> -<h3 id="org24573ef">About my Navi :</h3> -<div class="outline-text-3" id="text-org24573ef"> +<div id="outline-container-org9ec60d5" class="outline-3"> +<h3 id="org9ec60d5">About my Navi :</h3> +<div class="outline-text-3" id="text-org9ec60d5"> <p> My current setup is : </p> @@ -97,23 +97,23 @@ My GNUPG (GPG) public key <a href="./src/txt/pubkey.asc">./src/txt/pubkey.asc</a </div> </div> </div> -<div id="outline-container-orgfe3e5d1" class="outline-2"> -<h2 id="orgfe3e5d1">Sign my Guestbook (External website warning)</h2> -<div class="outline-text-2" id="text-orgfe3e5d1"> +<div id="outline-container-orgcc7a49e" class="outline-2"> +<h2 id="orgcc7a49e">Sign my Guestbook (External website warning)</h2> +<div class="outline-text-2" id="text-orgcc7a49e"> <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="orgd989cc4" class="figure"> +<div id="orgad3aec9" 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-org9bedce3" class="outline-2"> -<h2 id="org9bedce3">Blinkies</h2> -<div class="outline-text-2" id="text-org9bedce3"> +<div id="outline-container-org0bbd893" class="outline-2"> +<h2 id="org0bbd893">Blinkies</h2> +<div class="outline-text-2" id="text-org0bbd893"> <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"> @@ -131,27 +131,27 @@ Want to leave a message, opinion, review or a salty insult ? Be sure to Sign my <a href="https://partysepe13.neocities.org/"><img src="./src/gifs/blinkies/partysepe.png" alt="partysepe.png" /></a> </p> </div> -<div id="outline-container-org731e7c2" class="outline-3"> -<h3 id="org731e7c2">My banner</h3> -<div class="outline-text-3" id="text-org731e7c2"> +<div id="outline-container-org7b9c7ba" class="outline-3"> +<h3 id="org7b9c7ba">My banner</h3> +<div class="outline-text-3" id="text-org7b9c7ba"> <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="org63b9b8a" class="figure"> +<div id="orga76ef9e" class="figure"> <p><img src="./src/gifs/crystal-tilde.gif" alt="crystal-tilde.gif" /> </p> </div> </div> </div> </div> -<div id="outline-container-orgcd90488" class="outline-2"> -<h2 id="orgcd90488"><a href="https://crystal.tilde.institute/links.html">Close this website, txEn eht nepO.(JAVASCRIPT WARNING)!!</a></h2> +<div id="outline-container-org0602735" class="outline-2"> +<h2 id="org0602735"><a href="https://crystal.tilde.institute/links.html">Close this website, txEn eht nepO.(JAVASCRIPT WARNING)!!</a></h2> </div> -<div id="outline-container-orgffb3d1f" class="outline-2"> -<h2 id="orgffb3d1f">Misc :</h2> -<div class="outline-text-2" id="text-orgffb3d1f"> +<div id="outline-container-org96d9915" class="outline-2"> +<h2 id="org96d9915">Misc :</h2> +<div class="outline-text-2" id="text-org96d9915"> <ol class="org-ol"> <li><b><a href="./uni_notes/">My University notes</a></b></li> </ol> @@ -160,7 +160,7 @@ If you enjoyed my website, you could link me on your personal website using this </div> <div id="postamble" class="status"> <p class="author">Author: Crystal</p> -<p class="date">Created: 2023-11-01 Wed 20:09</p> +<p class="date">Created: 2023-11-01 Wed 20:15</p> </div> </body> </html> \ No newline at end of file diff --git a/links.html b/links.html index c29229f..1fcb993 100755 --- a/links.html +++ b/links.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <head> -<!-- 2023-11-01 Wed 20:09 --> +<!-- 2023-11-01 Wed 20:16 --> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <title>Close this website, txEn eht nepO.(JavaScript ahead) 💜</title> @@ -11,7 +11,7 @@ <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"/> -<link rel="icon" type="image/x-icon" href="favicon.ico"> +<link rel="icon" type="image/x-icon" href="favicon.png"> </head> <body> <div id="org-div-home-and-up"> @@ -26,9 +26,9 @@ </span> </h1> </div> -<div id="outline-container-org64c070d" class="outline-2"> -<h2 id="org64c070d">Webrings & Links</h2> -<div class="outline-text-2" id="text-org64c070d"> +<div id="outline-container-org1fe8c5b" class="outline-2"> +<h2 id="org1fe8c5b">Webrings & Links</h2> +<div class="outline-text-2" id="text-org1fe8c5b"> <p> <b>This site is a proud member of the geekring! Check some other geeky websites here!</b><br /> </p> @@ -65,9 +65,9 @@ href="https://teethinvitro.neocities.org/webring/linuxring/script/onionring.css" </tr> </table> </div> -<div id="outline-container-org5013547" class="outline-3"> -<h3 id="org5013547">Lainchan Webring</h3> -<div class="outline-text-3" id="text-org5013547"> +<div id="outline-container-org209b7a2" class="outline-3"> +<h3 id="org209b7a2">Lainchan Webring</h3> +<div class="outline-text-3" id="text-org209b7a2"> <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.<br /> </p> @@ -99,7 +99,7 @@ document.addEventListener("DOMContentLoaded", function(event) { </div> <div id="postamble" class="status"> <p class="author">Author: Crystal</p> -<p class="date">Created: 2023-11-01 Wed 20:09</p> +<p class="date">Created: 2023-11-01 Wed 20:16</p> </div> </body> </html> \ No newline at end of file diff --git a/src/org/articles/discord.org b/src/org/articles/discord.org index 500cfd9..7608f15 100755 --- a/src/org/articles/discord.org +++ b/src/org/articles/discord.org @@ -7,7 +7,7 @@ #+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../src/css/style.css"/> #+OPTIONS: html-style:nil #+OPTIONS: toc:nil -#+HTML_HEAD: <link rel="icon" type="image/x-icon" href="../../favicon.ico"> +#+HTML_HEAD: <link rel="icon" type="image/x-icon" href="../../favicon.png"> * Preamble diff --git a/src/org/index.org b/src/org/index.org index 439a055..90d5440 100755 --- a/src/org/index.org +++ b/src/org/index.org @@ -5,7 +5,7 @@ #+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"/> -#+HTML_HEAD: <link rel="icon" type="image/x-icon" href="favicon.ico"> +#+HTML_HEAD: <link rel="icon" type="image/x-icon" href="favicon.png"> #+OPTIONS: html-style:nil #+OPTIONS: toc:nil * Welcome to the wired diff --git a/src/org/links.org b/src/org/links.org index a1033e6..93f819d 100755 --- a/src/org/links.org +++ b/src/org/links.org @@ -11,7 +11,7 @@ #+OPTIONS: title:nil #+HTML_LINK_HOME: https://crystal.tilde.institute/ #+HTML_LINK_UP: https://crystal.tilde.institute/ -#+HTML_HEAD: <link rel="icon" type="image/x-icon" href="favicon.ico"> +#+HTML_HEAD: <link rel="icon" type="image/x-icon" href="favicon.png"> #+BEGIN_EXPORT html <div clas="glitch-container"> <h1 class="title glitch"> diff --git a/src/org/uni_notes/algebra1.org b/src/org/uni_notes/algebra1.org index 79ab1af..bc4f529 100755 --- a/src/org/uni_notes/algebra1.org +++ b/src/org/uni_notes/algebra1.org @@ -11,7 +11,7 @@ #+HTML_LINK_HOME: https://crystal.tilde.institute/ #+HTML_LINK_UP: ../../../uni_notes/ #+OPTIONS: tex:imagemagick -#+HTML_HEAD: <link rel="icon" type="image/x-icon" href="https://crystal.tilde.institute/favicon.ico"> +#+HTML_HEAD: <link rel="icon" type="image/x-icon" href="https://crystal.tilde.institute/favicon.png"> * Contenu de la Matiére ** Rappels et compléments (11H) - Logique mathématique et méthodes du raisonnement mathématique diff --git a/src/org/uni_notes/alsd1.org b/src/org/uni_notes/alsd1.org index 9b8e115..7f9397e 100755 --- a/src/org/uni_notes/alsd1.org +++ b/src/org/uni_notes/alsd1.org @@ -8,7 +8,7 @@ #+OPTIONS: html-style:nil #+OPTIONS: toc:4 #+HTML_LINK_HOME: https://crystal.tilde.institute/ -#+HTML_HEAD: <link rel="icon" type="image/x-icon" href="https://crystal.tilde.institute/favicon.ico"> +#+HTML_HEAD: <link rel="icon" type="image/x-icon" href="https://crystal.tilde.institute/favicon.png"> #+HTML_LINK_UP: ../../../uni_notes/ #+OPTIONS: \n:y * Contenu de la Matiére diff --git a/src/org/uni_notes/analyse1.org b/src/org/uni_notes/analyse1.org index 152046b..545ef2d 100755 --- a/src/org/uni_notes/analyse1.org +++ b/src/org/uni_notes/analyse1.org @@ -9,7 +9,7 @@ #+HTML_LINK_UP: ../../../uni_notes/ #+OPTIONS: html-style:nil #+OPTIONS: toc:4 -#+HTML_HEAD: <link rel="icon" type="image/x-icon" href="https://crystal.tilde.institute/favicon.ico"> +#+HTML_HEAD: <link rel="icon" type="image/x-icon" href="https://crystal.tilde.institute/favicon.png"> #+OPTIONS: \n:y * Contenu de la Matiére ** Chapitre 1 : Quelque propriétés de ℝ diff --git a/src/org/uni_notes/architecture1.org b/src/org/uni_notes/architecture1.org index 6c2d037..9891906 100755 --- a/src/org/uni_notes/architecture1.org +++ b/src/org/uni_notes/architecture1.org @@ -9,7 +9,7 @@ #+OPTIONS: toc:4 #+OPTIONS: \n:y #+HTML_LINK_HOME: https://crystal.tilde.institute/ -#+HTML_HEAD: <link rel="icon" type="image/x-icon" href="https://crystal.tilde.institute/favicon.ico"> +#+HTML_HEAD: <link rel="icon" type="image/x-icon" href="https://crystal.tilde.institute/favicon.png"> #+HTML_LINK_UP: ../../../uni_notes/ * Premier cours : Les systémes de numération /Sep 27/ : diff --git a/uni_notes/algebra.html b/uni_notes/algebra.html index c42a1b5..0e223c5 100755 --- a/uni_notes/algebra.html +++ b/uni_notes/algebra.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <head> -<!-- 2023-10-23 Mon 19:39 --> +<!-- 2023-11-01 Wed 20:17 --> <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> @@ -11,6 +11,7 @@ <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"/> +<link rel="icon" type="image/x-icon" href="https://crystal.tilde.institute/favicon.png"> </head> <body> <div id="org-div-home-and-up"> @@ -23,161 +24,161 @@ <h2>Table of Contents</h2> <div id="text-table-of-contents" role="doc-toc"> <ul> -<li><a href="#org4a250bc">Contenu de la Matiére</a> +<li><a href="#org42f27fc">Contenu de la Matiére</a> <ul> -<li><a href="#orga898532">Rappels et compléments (11H)</a></li> -<li><a href="#org01bcbce">Structures Algébriques (11H)</a></li> -<li><a href="#orgbe1e218">Polynômes et fractions rationnelles</a></li> +<li><a href="#orgf20cf94">Rappels et compléments (11H)</a></li> +<li><a href="#orgf700058">Structures Algébriques (11H)</a></li> +<li><a href="#org7a29a82">Polynômes et fractions rationnelles</a></li> </ul> </li> -<li><a href="#orgc75277f">Premier cours : Logique mathématique et méthodes du raisonnement mathématique <i>Sep 25</i> :</a> +<li><a href="#org7207cb0">Premier cours : Logique mathématique et méthodes du raisonnement mathématique <i>Sep 25</i> :</a> <ul> -<li><a href="#orge4d9578">Properties:</a> +<li><a href="#orgb936329">Properties:</a> <ul> -<li><a href="#orgb979f5e"><b>Absorption</b>:</a></li> -<li><a href="#org747a426"><b>Commutativity</b>:</a></li> -<li><a href="#org515acb7"><b>Associativity</b>:</a></li> -<li><a href="#orgbd31315"><b>Distributivity</b>:</a></li> -<li><a href="#org6ffa08f"><b>Neutral element</b>:</a></li> -<li><a href="#org5687242"><b>Negation of a conjunction & a disjunction</b>:</a></li> -<li><a href="#orge1d09c1"><b>Transitivity</b>:</a></li> -<li><a href="#orgcbc82d0"><b>Contraposition</b>:</a></li> -<li><a href="#org41b3b67">God only knows what this property is called:</a></li> +<li><a href="#orgf5da498"><b>Absorption</b>:</a></li> +<li><a href="#org49dbf9d"><b>Commutativity</b>:</a></li> +<li><a href="#orge255044"><b>Associativity</b>:</a></li> +<li><a href="#org31cc6c8"><b>Distributivity</b>:</a></li> +<li><a href="#orgf861930"><b>Neutral element</b>:</a></li> +<li><a href="#org8cb6e02"><b>Negation of a conjunction & a disjunction</b>:</a></li> +<li><a href="#orgfe01ac7"><b>Transitivity</b>:</a></li> +<li><a href="#org976f527"><b>Contraposition</b>:</a></li> +<li><a href="#org0865f2b">God only knows what this property is called:</a></li> </ul> </li> -<li><a href="#org8339e2b">Some exercices I found online :</a> +<li><a href="#org316b141">Some exercices I found online :</a> <ul> -<li><a href="#org4e164f4">USTHB 2022/2023 Section B :</a></li> +<li><a href="#orga3825f4">USTHB 2022/2023 Section B :</a></li> </ul> </li> </ul> </li> -<li><a href="#org966520a">2éme cours <i>Oct 2</i></a> +<li><a href="#org21d6c03">2éme cours <i>Oct 2</i></a> <ul> -<li><a href="#orgafe4a7b">Quantifiers</a> +<li><a href="#orgd6c9f49">Quantifiers</a> <ul> -<li><a href="#org5441c86">Proprieties</a></li> +<li><a href="#orgb332b43">Proprieties</a></li> </ul> </li> -<li><a href="#orga224095">Multi-parameter proprieties :</a></li> -<li><a href="#org019a7b1">Methods of mathematical reasoning :</a> +<li><a href="#orged685c1">Multi-parameter proprieties :</a></li> +<li><a href="#org78d7ed0">Methods of mathematical reasoning :</a> <ul> -<li><a href="#org235fea9">Direct reasoning :</a></li> -<li><a href="#orgae58c37">Reasoning by the Absurd:</a></li> -<li><a href="#orgb7e7f5f">Reasoning by contraposition:</a></li> -<li><a href="#org2199fd0">Reasoning by counter example:</a></li> +<li><a href="#org7d21c38">Direct reasoning :</a></li> +<li><a href="#orgcfd8723">Reasoning by the Absurd:</a></li> +<li><a href="#org102d3fa">Reasoning by contraposition:</a></li> +<li><a href="#org81cb388">Reasoning by counter example:</a></li> </ul> </li> </ul> </li> -<li><a href="#org97d10cc">3eme Cours : <i>Oct 9</i></a> +<li><a href="#orgc2178b8">3eme Cours : <i>Oct 9</i></a> <ul> <li> <ul> -<li><a href="#orgc324cd9">Reasoning by recurrence :</a></li> +<li><a href="#org4855f6f">Reasoning by recurrence :</a></li> </ul> </li> </ul> </li> -<li><a href="#org924a736">4eme Cours : Chapitre 2 : Sets and Operations</a> +<li><a href="#orgde6bfac">4eme Cours : Chapitre 2 : Sets and Operations</a> <ul> -<li><a href="#org9d2cb2d">Definition of a set :</a></li> -<li><a href="#org3b107a4">Belonging, inclusion, and equality :</a></li> -<li><a href="#orgbba2f2c">Intersections and reunions :</a> +<li><a href="#orgfe8000a">Definition of a set :</a></li> +<li><a href="#orgfe04671">Belonging, inclusion, and equality :</a></li> +<li><a href="#orga2eb99d">Intersections and reunions :</a> <ul> -<li><a href="#org4fd2e61">Intersection:</a></li> -<li><a href="#orgabd991c">Union:</a></li> -<li><a href="#org083102e">Difference between two sets:</a></li> -<li><a href="#orgcc1969e">Complimentary set:</a></li> -<li><a href="#org72511ff">Symmetrical difference</a></li> +<li><a href="#org560d563">Intersection:</a></li> +<li><a href="#org7147bc3">Union:</a></li> +<li><a href="#org16b5ab2">Difference between two sets:</a></li> +<li><a href="#orgdac190b">Complimentary set:</a></li> +<li><a href="#org4e0b111">Symmetrical difference</a></li> </ul> </li> -<li><a href="#org44b1b96">Proprieties :</a> +<li><a href="#org691c863">Proprieties :</a> <ul> -<li><a href="#orga3eac79">Commutativity:</a></li> -<li><a href="#org1a9121a">Associativity:</a></li> -<li><a href="#orgaf7fe3b">Distributivity:</a></li> -<li><a href="#org658b728">Lois de Morgan:</a></li> -<li><a href="#org4f0dd58">An other one:</a></li> -<li><a href="#orgdd60033">An other one:</a></li> -<li><a href="#orgbf5feb1">And an other one:</a></li> -<li><a href="#orgefabd47">And the last one:</a></li> +<li><a href="#org9cc9f31">Commutativity:</a></li> +<li><a href="#org471083b">Associativity:</a></li> +<li><a href="#orge63be10">Distributivity:</a></li> +<li><a href="#orgfb01947">Lois de Morgan:</a></li> +<li><a href="#orge1a41eb">An other one:</a></li> +<li><a href="#org9939b0f">An other one:</a></li> +<li><a href="#org90bfdc4">And an other one:</a></li> +<li><a href="#org1e49001">And the last one:</a></li> </ul> </li> </ul> </li> -<li><a href="#orgc9ed06c">5eme cours: L’ensemble des parties d’un ensemble <i>Oct 16</i></a> +<li><a href="#org272eca3">5eme cours: L’ensemble des parties d’un ensemble <i>Oct 16</i></a> <ul> <li> <ul> -<li><a href="#org7a1da3c">Notes :</a></li> -<li><a href="#orga6e5f8a">Examples :</a></li> +<li><a href="#org5b29e32">Notes :</a></li> +<li><a href="#org5636bd8">Examples :</a></li> </ul> </li> -<li><a href="#org7286ec5">Partition of a set :</a></li> -<li><a href="#orgd8e00ac">Cartesian products :</a> +<li><a href="#orgd8cb2c3">Partition of a set :</a></li> +<li><a href="#orgf40404d">Cartesian products :</a> <ul> -<li><a href="#orgdb491ee">Example :</a></li> -<li><a href="#org28c23b2">Some proprieties:</a></li> +<li><a href="#orgd526cb8">Example :</a></li> +<li><a href="#org56dd088">Some proprieties:</a></li> </ul> </li> </ul> </li> -<li><a href="#orgfdbe4f3">Binary relations in a set :</a> +<li><a href="#org5ee4278">Binary relations in a set :</a> <ul> -<li><a href="#org3696656">Definition :</a></li> -<li><a href="#orgd32f673">Proprieties :</a></li> -<li><a href="#org22f460a">Equivalence relationship :</a> +<li><a href="#orgddc9af6">Definition :</a></li> +<li><a href="#orge65424e">Proprieties :</a></li> +<li><a href="#orgd7877d3">Equivalence relationship :</a> <ul> -<li><a href="#org68ddde2">Equivalence class :</a></li> +<li><a href="#org85cf025">Equivalence class :</a></li> </ul> </li> -<li><a href="#orge976c7e">Order relationship :</a> +<li><a href="#orge18dcc7">Order relationship :</a> <ul> -<li><a href="#org1f19847"><span class="todo TODO">TODO</span> Examples :</a></li> +<li><a href="#org60d471a"><span class="todo TODO">TODO</span> Examples :</a></li> </ul> </li> </ul> </li> -<li><a href="#orgc956a73">TP exercices <i>Oct 20</i> :</a> +<li><a href="#org77de6e3">TP exercices <i>Oct 20</i> :</a> <ul> -<li><a href="#org15b0e75">Exercice 3 :</a> +<li><a href="#org3ca8006">Exercice 3 :</a> <ul> -<li><a href="#orgb132892">Question 3</a></li> +<li><a href="#orgad95ec3">Question 3</a></li> </ul> </li> -<li><a href="#org9a4006b">Exercice 4 :</a> +<li><a href="#org8180ae0">Exercice 4 :</a> <ul> -<li><a href="#org43cf6d6"><span class="done DONE">DONE</span> Question 1 :</a></li> +<li><a href="#orgfe0b1e2"><span class="done DONE">DONE</span> Question 1 :</a></li> </ul> </li> </ul> </li> -<li><a href="#org429ab91">Chapter 3 : Applications</a> +<li><a href="#org2d6e0ba">Chapter 3 : Applications</a> <ul> -<li><a href="#org4a4b3cf">3.1 Generalities about applications :</a> +<li><a href="#orga5be12f">3.1 Generalities about applications :</a> <ul> -<li><a href="#org0cac0c6">Definition :</a></li> -<li><a href="#orgc048e93">Restriction and prolongation of an application :</a></li> -<li><a href="#org8e61361">Composition of applications :</a></li> +<li><a href="#org805d7bc">Definition :</a></li> +<li><a href="#org7947331">Restriction and prolongation of an application :</a></li> +<li><a href="#orgd94bc69">Composition of applications :</a></li> </ul> </li> -<li><a href="#org5c096db">3.2 Injection, surjection and bijection :</a> +<li><a href="#org257d05a">3.2 Injection, surjection and bijection :</a> <ul> -<li><a href="#org4162b56">Proposition :</a></li> +<li><a href="#org1612e09">Proposition :</a></li> </ul> </li> -<li><a href="#org736de6c">3.3 Reciprocal applications :</a> +<li><a href="#orgebdf518">3.3 Reciprocal applications :</a> <ul> -<li><a href="#orgafb7f85">Def :</a></li> -<li><a href="#orgec3a3d6">Theorem :</a></li> -<li><a href="#org940e2d6">Some proprieties :</a></li> +<li><a href="#orgf072e42">Def :</a></li> +<li><a href="#org244b352">Theorem :</a></li> +<li><a href="#org1479c0e">Some proprieties :</a></li> </ul> </li> -<li><a href="#org2d173c2">3.4 Direct Image and reciprocal Image :</a> +<li><a href="#orgaf81bb3">3.4 Direct Image and reciprocal Image :</a> <ul> -<li><a href="#org769c809">Direct Image :</a></li> -<li><a href="#org7d705d3">Reciprocal image :</a></li> +<li><a href="#org87b91e2">Direct Image :</a></li> +<li><a href="#org500bc40">Reciprocal image :</a></li> </ul> </li> </ul> @@ -185,13 +186,13 @@ </ul> </div> </div> -<div id="outline-container-org4a250bc" class="outline-2"> -<h2 id="org4a250bc">Contenu de la Matiére</h2> -<div class="outline-text-2" id="text-org4a250bc"> +<div id="outline-container-org42f27fc" class="outline-2"> +<h2 id="org42f27fc">Contenu de la Matiére</h2> +<div class="outline-text-2" id="text-org42f27fc"> </div> -<div id="outline-container-orga898532" class="outline-3"> -<h3 id="orga898532">Rappels et compléments (11H)</h3> -<div class="outline-text-3" id="text-orga898532"> +<div id="outline-container-orgf20cf94" class="outline-3"> +<h3 id="orgf20cf94">Rappels et compléments (11H)</h3> +<div class="outline-text-3" id="text-orgf20cf94"> <ul class="org-ul"> <li>Logique mathématique et méthodes du raisonnement mathématique<br /></li> <li>Ensembles et Relations<br /></li> @@ -199,9 +200,9 @@ </ul> </div> </div> -<div id="outline-container-org01bcbce" class="outline-3"> -<h3 id="org01bcbce">Structures Algébriques (11H)</h3> -<div class="outline-text-3" id="text-org01bcbce"> +<div id="outline-container-orgf700058" class="outline-3"> +<h3 id="orgf700058">Structures Algébriques (11H)</h3> +<div class="outline-text-3" id="text-orgf700058"> <ul class="org-ul"> <li>Groupes et morphisme de groupes<br /></li> <li>Anneaux et morphisme d’anneaux<br /></li> @@ -209,9 +210,9 @@ </ul> </div> </div> -<div id="outline-container-orgbe1e218" class="outline-3"> -<h3 id="orgbe1e218">Polynômes et fractions rationnelles</h3> -<div class="outline-text-3" id="text-orgbe1e218"> +<div id="outline-container-org7a29a82" class="outline-3"> +<h3 id="org7a29a82">Polynômes et fractions rationnelles</h3> +<div class="outline-text-3" id="text-org7a29a82"> <ul class="org-ul"> <li>Notion du polynôme à une indéterminée á coefficients dans un anneau<br /></li> <li>Opérations Algébriques sur les polynômes<br /></li> @@ -224,9 +225,9 @@ </div> </div> </div> -<div id="outline-container-orgc75277f" class="outline-2"> -<h2 id="orgc75277f">Premier cours : Logique mathématique et méthodes du raisonnement mathématique <i>Sep 25</i> :</h2> -<div class="outline-text-2" id="text-orgc75277f"> +<div id="outline-container-org7207cb0" class="outline-2"> +<h2 id="org7207cb0">Premier cours : Logique mathématique et méthodes du raisonnement mathématique <i>Sep 25</i> :</h2> +<div class="outline-text-2" id="text-org7207cb0"> <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.<br /> </p> @@ -576,13 +577,13 @@ A proposition is equivalent to another only when both of them have <b>the same v <i>Note: P implying Q is equivalent to P̅ implying Q̅, or: (P ⇒ Q) ⇔ (P̅ ⇒ Q̅)</i><br /> </p> </div> -<div id="outline-container-orge4d9578" class="outline-3"> -<h3 id="orge4d9578">Properties:</h3> -<div class="outline-text-3" id="text-orge4d9578"> +<div id="outline-container-orgb936329" class="outline-3"> +<h3 id="orgb936329">Properties:</h3> +<div class="outline-text-3" id="text-orgb936329"> </div> -<div id="outline-container-orgb979f5e" class="outline-4"> -<h4 id="orgb979f5e"><b>Absorption</b>:</h4> -<div class="outline-text-4" id="text-orgb979f5e"> +<div id="outline-container-orgf5da498" class="outline-4"> +<h4 id="orgf5da498"><b>Absorption</b>:</h4> +<div class="outline-text-4" id="text-orgf5da498"> <p> (P ∨ P) ⇔ P<br /> </p> @@ -592,9 +593,9 @@ A proposition is equivalent to another only when both of them have <b>the same v </p> </div> </div> -<div id="outline-container-org747a426" class="outline-4"> -<h4 id="org747a426"><b>Commutativity</b>:</h4> -<div class="outline-text-4" id="text-org747a426"> +<div id="outline-container-org49dbf9d" class="outline-4"> +<h4 id="org49dbf9d"><b>Commutativity</b>:</h4> +<div class="outline-text-4" id="text-org49dbf9d"> <p> (P ∧ Q) ⇔ (Q ∧ P)<br /> </p> @@ -604,9 +605,9 @@ A proposition is equivalent to another only when both of them have <b>the same v </p> </div> </div> -<div id="outline-container-org515acb7" class="outline-4"> -<h4 id="org515acb7"><b>Associativity</b>:</h4> -<div class="outline-text-4" id="text-org515acb7"> +<div id="outline-container-orge255044" class="outline-4"> +<h4 id="orge255044"><b>Associativity</b>:</h4> +<div class="outline-text-4" id="text-orge255044"> <p> P ∧ (Q ∧ R) ⇔ (P ∧ Q) ∧ R<br /> </p> @@ -616,9 +617,9 @@ P ∨ (Q ∨ R) ⇔ (P ∨ Q) ∨ R<br /> </p> </div> </div> -<div id="outline-container-orgbd31315" class="outline-4"> -<h4 id="orgbd31315"><b>Distributivity</b>:</h4> -<div class="outline-text-4" id="text-orgbd31315"> +<div id="outline-container-org31cc6c8" class="outline-4"> +<h4 id="org31cc6c8"><b>Distributivity</b>:</h4> +<div class="outline-text-4" id="text-org31cc6c8"> <p> P ∧ (Q ∨ R) ⇔ (P ∧ Q) ∨ (P ∧ R)<br /> </p> @@ -628,9 +629,9 @@ P ∨ (Q ∧ R) ⇔ (P ∨ Q) ∧ (P ∨ R)<br /> </p> </div> </div> -<div id="outline-container-org6ffa08f" class="outline-4"> -<h4 id="org6ffa08f"><b>Neutral element</b>:</h4> -<div class="outline-text-4" id="text-org6ffa08f"> +<div id="outline-container-orgf861930" class="outline-4"> +<h4 id="orgf861930"><b>Neutral element</b>:</h4> +<div class="outline-text-4" id="text-orgf861930"> <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><br /> </p> @@ -644,9 +645,9 @@ P ∨ F ⇔ P<br /> </p> </div> </div> -<div id="outline-container-org5687242" class="outline-4"> -<h4 id="org5687242"><b>Negation of a conjunction & a disjunction</b>:</h4> -<div class="outline-text-4" id="text-org5687242"> +<div id="outline-container-org8cb6e02" class="outline-4"> +<h4 id="org8cb6e02"><b>Negation of a conjunction & a disjunction</b>:</h4> +<div class="outline-text-4" id="text-org8cb6e02"> <p> Now we won’t use bars here because my lazy ass doesn’t know how, so instead I will use not()!!!<br /> </p> @@ -664,25 +665,25 @@ not(<b>P ∨ Q</b>) ⇔ P̅ ∧ Q̅<br /> </p> </div> </div> -<div id="outline-container-orge1d09c1" class="outline-4"> -<h4 id="orge1d09c1"><b>Transitivity</b>:</h4> -<div class="outline-text-4" id="text-orge1d09c1"> +<div id="outline-container-orgfe01ac7" class="outline-4"> +<h4 id="orgfe01ac7"><b>Transitivity</b>:</h4> +<div class="outline-text-4" id="text-orgfe01ac7"> <p> [(P ⇒ Q) AND (Q ⇒ R)] ⇔ P ⇒ R<br /> </p> </div> </div> -<div id="outline-container-orgcbc82d0" class="outline-4"> -<h4 id="orgcbc82d0"><b>Contraposition</b>:</h4> -<div class="outline-text-4" id="text-orgcbc82d0"> +<div id="outline-container-org976f527" class="outline-4"> +<h4 id="org976f527"><b>Contraposition</b>:</h4> +<div class="outline-text-4" id="text-org976f527"> <p> (P ⇒ Q) ⇔ (Q̅ ⇒ P̅)<br /> </p> </div> </div> -<div id="outline-container-org41b3b67" class="outline-4"> -<h4 id="org41b3b67">God only knows what this property is called:</h4> -<div class="outline-text-4" id="text-org41b3b67"> +<div id="outline-container-org0865f2b" class="outline-4"> +<h4 id="org0865f2b">God only knows what this property is called:</h4> +<div class="outline-text-4" id="text-org0865f2b"> <p> <i>If</i><br /> </p> @@ -709,17 +710,17 @@ Q is always true<br /> </div> </div> </div> -<div id="outline-container-org8339e2b" class="outline-3"> -<h3 id="org8339e2b">Some exercices I found online :</h3> -<div class="outline-text-3" id="text-org8339e2b"> +<div id="outline-container-org316b141" class="outline-3"> +<h3 id="org316b141">Some exercices I found online :</h3> +<div class="outline-text-3" id="text-org316b141"> </div> -<div id="outline-container-org4e164f4" class="outline-4"> -<h4 id="org4e164f4">USTHB 2022/2023 Section B :</h4> -<div class="outline-text-4" id="text-org4e164f4"> +<div id="outline-container-orga3825f4" class="outline-4"> +<h4 id="orga3825f4">USTHB 2022/2023 Section B :</h4> +<div class="outline-text-4" id="text-orga3825f4"> </div> <ul class="org-ul"> -<li><a id="org1202cc7"></a>Exercice 1: Démontrer les équivalences suivantes:<br /> -<div class="outline-text-5" id="text-org1202cc7"> +<li><a id="orge27aa8d"></a>Exercice 1: Démontrer les équivalences suivantes:<br /> +<div class="outline-text-5" id="text-orge27aa8d"> <ol class="org-ol"> <li><p> (P ⇒ Q) ⇔ (Q̅ ⇒ P̅)<br /> @@ -773,8 +774,8 @@ Literally the same as above 🩷<br /> </ol> </div> </li> -<li><a id="org993b830"></a>Exercice 2: Dire si les propositions suivantes sont vraies ou fausses, et les nier:<br /> -<div class="outline-text-5" id="text-org993b830"> +<li><a id="orgd9c7023"></a>Exercice 2: Dire si les propositions suivantes sont vraies ou fausses, et les nier:<br /> +<div class="outline-text-5" id="text-orgd9c7023"> <ol class="org-ol"> <li><p> ∀x ∈ ℝ ,∃y ∈ ℝ*+, tels que e^x = y<br /> @@ -907,13 +908,13 @@ y + x < 8<br /> </div> </div> </div> -<div id="outline-container-org966520a" class="outline-2"> -<h2 id="org966520a">2éme cours <i>Oct 2</i></h2> -<div class="outline-text-2" id="text-org966520a"> +<div id="outline-container-org21d6c03" class="outline-2"> +<h2 id="org21d6c03">2éme cours <i>Oct 2</i></h2> +<div class="outline-text-2" id="text-org21d6c03"> </div> -<div id="outline-container-orgafe4a7b" class="outline-3"> -<h3 id="orgafe4a7b">Quantifiers</h3> -<div class="outline-text-3" id="text-orgafe4a7b"> +<div id="outline-container-orgd6c9f49" class="outline-3"> +<h3 id="orgd6c9f49">Quantifiers</h3> +<div class="outline-text-3" id="text-orgd6c9f49"> <p> A propriety P can depend on a parameter x<br /> </p> @@ -929,8 +930,8 @@ A propriety P can depend on a parameter x<br /> </p> </div> <ul class="org-ul"> -<li><a id="org11a2eef"></a>Example<br /> -<div class="outline-text-6" id="text-org11a2eef"> +<li><a id="orge92d880"></a>Example<br /> +<div class="outline-text-6" id="text-orge92d880"> <p> P(x) : x+1≥0<br /> </p> @@ -941,13 +942,13 @@ P(X) is True or False depending on the values of x<br /> </div> </li> </ul> -<div id="outline-container-org5441c86" class="outline-4"> -<h4 id="org5441c86">Proprieties</h4> -<div class="outline-text-4" id="text-org5441c86"> +<div id="outline-container-orgb332b43" class="outline-4"> +<h4 id="orgb332b43">Proprieties</h4> +<div class="outline-text-4" id="text-orgb332b43"> </div> <ul class="org-ul"> -<li><a id="org92d20b7"></a>Propriety Number 1:<br /> -<div class="outline-text-5" id="text-org92d20b7"> +<li><a id="org8587885"></a>Propriety Number 1:<br /> +<div class="outline-text-5" id="text-org8587885"> <p> The negation of the universal quantifier is the existential quantifier, and vice-versa :<br /> </p> @@ -958,8 +959,8 @@ The negation of the universal quantifier is the existential quantifier, and vice </ul> </div> <ul class="org-ul"> -<li><a id="orgd155b7d"></a>Example:<br /> -<div class="outline-text-6" id="text-orgd155b7d"> +<li><a id="org3a19f5f"></a>Example:<br /> +<div class="outline-text-6" id="text-org3a19f5f"> <p> ∀ x ≥ 1 x² > 5 ⇔ ∃ x ≥ 1 x² < 5<br /> </p> @@ -967,8 +968,8 @@ The negation of the universal quantifier is the existential quantifier, and vice </li> </ul> </li> -<li><a id="orga5c524a"></a>Propriety Number 2:<br /> -<div class="outline-text-5" id="text-orga5c524a"> +<li><a id="orgab7b647"></a>Propriety Number 2:<br /> +<div class="outline-text-5" id="text-orgab7b647"> <p> <b>∀x ∈ E, [P(x) ∧ Q(x)] ⇔ [∀ x ∈ E, P(x)] ∧ [∀ x ∈ E, Q(x)]</b><br /> </p> @@ -979,8 +980,8 @@ The propriety “For any value of x from a set E , P(x) and Q(x)” is e </p> </div> <ul class="org-ul"> -<li><a id="orgfbd033f"></a>Example :<br /> -<div class="outline-text-6" id="text-orgfbd033f"> +<li><a id="org8ba49ff"></a>Example :<br /> +<div class="outline-text-6" id="text-org8ba49ff"> <p> P(x) : sqrt(x) > 0 ; Q(x) : x ≥ 1<br /> </p> @@ -998,8 +999,8 @@ P(x) : sqrt(x) > 0 ; Q(x) : x ≥ 1<br /> </li> </ul> </li> -<li><a id="orgf25f95e"></a>Propriety Number 3:<br /> -<div class="outline-text-5" id="text-orgf25f95e"> +<li><a id="org91796f9"></a>Propriety Number 3:<br /> +<div class="outline-text-5" id="text-org91796f9"> <p> <b>∃ x ∈ E, [P(x) ∧ Q(x)] <i>⇒</i> [∃ x ∈ E, P(x)] ∧ [∃ x ∈ E, Q(x)]</b><br /> </p> @@ -1010,8 +1011,8 @@ P(x) : sqrt(x) > 0 ; Q(x) : x ≥ 1<br /> </p> </div> <ul class="org-ul"> -<li><a id="org1eb9ad5"></a>Example of why it’s NOT an equivalence :<br /> -<div class="outline-text-6" id="text-org1eb9ad5"> +<li><a id="org1f20a27"></a>Example of why it’s NOT an equivalence :<br /> +<div class="outline-text-6" id="text-org1f20a27"> <p> P(x) : x > 5 ; Q(x) : x < 5<br /> </p> @@ -1024,8 +1025,8 @@ Of course there is no value of x such as its inferior and superior to 5 at the s </li> </ul> </li> -<li><a id="org1b17eb0"></a>Propriety Number 4:<br /> -<div class="outline-text-5" id="text-org1b17eb0"> +<li><a id="org2b9f54b"></a>Propriety Number 4:<br /> +<div class="outline-text-5" id="text-org2b9f54b"> <p> <b>[∀ x ∈ E, P(x)] ∨ [∀ x ∈ E, Q(x)] <i>⇒</i> ∀x ∈ E, [P(x) ∨ Q(x)]</b><br /> </p> @@ -1039,16 +1040,16 @@ Of course there is no value of x such as its inferior and superior to 5 at the s </ul> </div> </div> -<div id="outline-container-orga224095" class="outline-3"> -<h3 id="orga224095">Multi-parameter proprieties :</h3> -<div class="outline-text-3" id="text-orga224095"> +<div id="outline-container-orged685c1" class="outline-3"> +<h3 id="orged685c1">Multi-parameter proprieties :</h3> +<div class="outline-text-3" id="text-orged685c1"> <p> A propriety P can depend on two or more parameters, for convenience we call them x,y,z…etc<br /> </p> </div> <ul class="org-ul"> -<li><a id="orgf04d41f"></a>Example :<br /> -<div class="outline-text-6" id="text-orgf04d41f"> +<li><a id="org747b217"></a>Example :<br /> +<div class="outline-text-6" id="text-org747b217"> <p> P(x,y): x+y > 0<br /> </p> @@ -1064,8 +1065,8 @@ P(-2,-1) is a False one<br /> </p> </div> </li> -<li><a id="orgb4df659"></a>WARNING :<br /> -<div class="outline-text-6" id="text-orgb4df659"> +<li><a id="org5d93eaf"></a>WARNING :<br /> +<div class="outline-text-6" id="text-org5d93eaf"> <p> ∀x ∈ E, ∃y ∈ F , P(x,y)<br /> </p> @@ -1081,8 +1082,8 @@ Are different because in the first one y depends on x, while in the second one, </p> </div> <ul class="org-ul"> -<li><a id="orge473c22"></a>Example :<br /> -<div class="outline-text-7" id="text-orge473c22"> +<li><a id="orgc60c61d"></a>Example :<br /> +<div class="outline-text-7" id="text-orgc60c61d"> <p> ∀ x ∈ ℕ , ∃ y ∈ ℕ y > x -–— True<br /> </p> @@ -1096,8 +1097,8 @@ Are different because in the first one y depends on x, while in the second one, </ul> </li> </ul> -<li><a id="org5f7adfc"></a>Proprieties :<br /> -<div class="outline-text-5" id="text-org5f7adfc"> +<li><a id="orgda9f614"></a>Proprieties :<br /> +<div class="outline-text-5" id="text-orgda9f614"> <ol class="org-ol"> <li>not(∀x ∈ E ,∃y ∈ F P(x,y)) ⇔ ∃x ∈ E, ∀y ∈ F not(P(x,y))<br /></li> <li>not(∃x ∈ E ,∀y ∈ F P(x,y)) ⇔ ∀x ∈ E, ∃y ∈ F not(P(x,y))<br /></li> @@ -1106,20 +1107,20 @@ Are different because in the first one y depends on x, while in the second one, </li> </ul> </div> -<div id="outline-container-org019a7b1" class="outline-3"> -<h3 id="org019a7b1">Methods of mathematical reasoning :</h3> -<div class="outline-text-3" id="text-org019a7b1"> +<div id="outline-container-org78d7ed0" class="outline-3"> +<h3 id="org78d7ed0">Methods of mathematical reasoning :</h3> +<div class="outline-text-3" id="text-org78d7ed0"> </div> -<div id="outline-container-org235fea9" class="outline-4"> -<h4 id="org235fea9">Direct reasoning :</h4> -<div class="outline-text-4" id="text-org235fea9"> +<div id="outline-container-org7d21c38" class="outline-4"> +<h4 id="org7d21c38">Direct reasoning :</h4> +<div class="outline-text-4" id="text-org7d21c38"> <p> To show that an implication P ⇒ Q is true, we suppose that P is true and we show that Q is true<br /> </p> </div> <ul class="org-ul"> -<li><a id="orgced11f5"></a>Example:<br /> -<div class="outline-text-5" id="text-orgced11f5"> +<li><a id="org59d34b3"></a>Example:<br /> +<div class="outline-text-5" id="text-org59d34b3"> <p> Let a,b be two Real numbers, we have to prove that <b>a² + b² = 1 ⇒ |a + b| ≤ 2</b><br /> </p> @@ -1162,9 +1163,9 @@ a²+b²=1 ⇒ |a + b| ≤ 2 <b>Which is what we wanted to prove, therefor the im </li> </ul> </div> -<div id="outline-container-orgae58c37" class="outline-4"> -<h4 id="orgae58c37">Reasoning by the Absurd:</h4> -<div class="outline-text-4" id="text-orgae58c37"> +<div id="outline-container-orgcfd8723" class="outline-4"> +<h4 id="orgcfd8723">Reasoning by the Absurd:</h4> +<div class="outline-text-4" id="text-orgcfd8723"> <p> To prove that a proposition is True, we suppose that it’s False and we must come to a contradiction<br /> </p> @@ -1175,8 +1176,8 @@ And to prove that an implication P ⇒ Q is true using the reasoning by the absu </p> </div> <ul class="org-ul"> -<li><a id="org8dc4906"></a>Example:<br /> -<div class="outline-text-5" id="text-org8dc4906"> +<li><a id="orga4a0e2d"></a>Example:<br /> +<div class="outline-text-5" id="text-orga4a0e2d"> <p> Prove that this proposition is correct using the reasoning by the absurd : ∀x ∈ ℝ* , sqrt(1+x²) ≠ 1 + x²/2<br /> </p> @@ -1194,17 +1195,17 @@ sqrt(1+x²) = 1 + x²/2 ; 1 + x² = (1+x²/2)² ; 1 + x² = 1 + x^4/4 + x² ; </li> </ul> </div> -<div id="outline-container-orgb7e7f5f" class="outline-4"> -<h4 id="orgb7e7f5f">Reasoning by contraposition:</h4> -<div class="outline-text-4" id="text-orgb7e7f5f"> +<div id="outline-container-org102d3fa" class="outline-4"> +<h4 id="org102d3fa">Reasoning by contraposition:</h4> +<div class="outline-text-4" id="text-org102d3fa"> <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<br /> </p> </div> </div> -<div id="outline-container-org2199fd0" class="outline-4"> -<h4 id="org2199fd0">Reasoning by counter example:</h4> -<div class="outline-text-4" id="text-org2199fd0"> +<div id="outline-container-org81cb388" class="outline-4"> +<h4 id="org81cb388">Reasoning by counter example:</h4> +<div class="outline-text-4" id="text-org81cb388"> <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<br /> </p> @@ -1212,20 +1213,20 @@ To prove that a proposition ∀x ∈ E, P(x) is false, all we have to do is find </div> </div> </div> -<div id="outline-container-org97d10cc" class="outline-2"> -<h2 id="org97d10cc">3eme Cours : <i>Oct 9</i></h2> -<div class="outline-text-2" id="text-org97d10cc"> +<div id="outline-container-orgc2178b8" class="outline-2"> +<h2 id="orgc2178b8">3eme Cours : <i>Oct 9</i></h2> +<div class="outline-text-2" id="text-orgc2178b8"> </div> -<div id="outline-container-orgc324cd9" class="outline-4"> -<h4 id="orgc324cd9">Reasoning by recurrence :</h4> -<div class="outline-text-4" id="text-orgc324cd9"> +<div id="outline-container-org4855f6f" class="outline-4"> +<h4 id="org4855f6f">Reasoning by recurrence :</h4> +<div class="outline-text-4" id="text-org4855f6f"> <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<br /> </p> </div> <ul class="org-ul"> -<li><a id="org00a2b7b"></a>Example:<br /> -<div class="outline-text-5" id="text-org00a2b7b"> +<li><a id="orga792d9c"></a>Example:<br /> +<div class="outline-text-5" id="text-orga792d9c"> <p> Let’s prove that ∀ n ≥ 1 , (n,k=1)Σk = [n(n+1)]/2<br /> </p> @@ -1261,21 +1262,21 @@ For n ≥ 1. We assume that P(n) is true, OR : <b>(n, k=1)Σk = n(n+1)/2</b>. We </ul> </div> </div> -<div id="outline-container-org924a736" class="outline-2"> -<h2 id="org924a736">4eme Cours : Chapitre 2 : Sets and Operations</h2> -<div class="outline-text-2" id="text-org924a736"> +<div id="outline-container-orgde6bfac" class="outline-2"> +<h2 id="orgde6bfac">4eme Cours : Chapitre 2 : Sets and Operations</h2> +<div class="outline-text-2" id="text-orgde6bfac"> </div> -<div id="outline-container-org9d2cb2d" class="outline-3"> -<h3 id="org9d2cb2d">Definition of a set :</h3> -<div class="outline-text-3" id="text-org9d2cb2d"> +<div id="outline-container-orgfe8000a" class="outline-3"> +<h3 id="orgfe8000a">Definition of a set :</h3> +<div class="outline-text-3" id="text-orgfe8000a"> <p> A set is a collection of objects that share the sane propriety<br /> </p> </div> </div> -<div id="outline-container-org3b107a4" class="outline-3"> -<h3 id="org3b107a4">Belonging, inclusion, and equality :</h3> -<div class="outline-text-3" id="text-org3b107a4"> +<div id="outline-container-orgfe04671" class="outline-3"> +<h3 id="orgfe04671">Belonging, inclusion, and equality :</h3> +<div class="outline-text-3" id="text-orgfe04671"> <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><br /></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><br /></li> @@ -1284,13 +1285,13 @@ A set is a collection of objects that share the sane propriety<br /> </ol> </div> </div> -<div id="outline-container-orgbba2f2c" class="outline-3"> -<h3 id="orgbba2f2c">Intersections and reunions :</h3> -<div class="outline-text-3" id="text-orgbba2f2c"> +<div id="outline-container-orga2eb99d" class="outline-3"> +<h3 id="orga2eb99d">Intersections and reunions :</h3> +<div class="outline-text-3" id="text-orga2eb99d"> </div> -<div id="outline-container-org4fd2e61" class="outline-4"> -<h4 id="org4fd2e61">Intersection:</h4> -<div class="outline-text-4" id="text-org4fd2e61"> +<div id="outline-container-org560d563" class="outline-4"> +<h4 id="org560d563">Intersection:</h4> +<div class="outline-text-4" id="text-org560d563"> <p> E ∩ F = {x / x ∈ E AND x ∈ F} ; x ∈ E ∩ F ⇔ x ∈ F AND x ∈ F<br /> </p> @@ -1301,9 +1302,9 @@ x ∉ E ∩ F ⇔ x ∉ E OR x ∉ F<br /> </p> </div> </div> -<div id="outline-container-orgabd991c" class="outline-4"> -<h4 id="orgabd991c">Union:</h4> -<div class="outline-text-4" id="text-orgabd991c"> +<div id="outline-container-org7147bc3" class="outline-4"> +<h4 id="org7147bc3">Union:</h4> +<div class="outline-text-4" id="text-org7147bc3"> <p> E ∪ F = {x / x ∈ E OR x ∈ F} ; x ∈ E ∪ F ⇔ x ∈ F OR x ∈ F<br /> </p> @@ -1314,17 +1315,17 @@ x ∉ E ∪ F ⇔ x ∉ E AND x ∉ F<br /> </p> </div> </div> -<div id="outline-container-org083102e" class="outline-4"> -<h4 id="org083102e">Difference between two sets:</h4> -<div class="outline-text-4" id="text-org083102e"> +<div id="outline-container-org16b5ab2" class="outline-4"> +<h4 id="org16b5ab2">Difference between two sets:</h4> +<div class="outline-text-4" id="text-org16b5ab2"> <p> E(Which is also written as : E - F) = {x / x ∈ E and x ∉ F}<br /> </p> </div> </div> -<div id="outline-container-orgcc1969e" class="outline-4"> -<h4 id="orgcc1969e">Complimentary set:</h4> -<div class="outline-text-4" id="text-orgcc1969e"> +<div id="outline-container-orgdac190b" class="outline-4"> +<h4 id="orgdac190b">Complimentary set:</h4> +<div class="outline-text-4" id="text-orgdac190b"> <p> If F ⊂ E. E - F is the complimentary of F in E.<br /> </p> @@ -1335,52 +1336,52 @@ FCE = {x /x ∈ E AND x ∉ F} <b>ONLY WHEN F IS A SUBSET OF E</b><br /> </p> </div> </div> -<div id="outline-container-org72511ff" class="outline-4"> -<h4 id="org72511ff">Symmetrical difference</h4> -<div class="outline-text-4" id="text-org72511ff"> +<div id="outline-container-org4e0b111" class="outline-4"> +<h4 id="org4e0b111">Symmetrical difference</h4> +<div class="outline-text-4" id="text-org4e0b111"> <p> E Δ F = (E - F) ∪ (F - E) ; = (E ∪ F) - (E ∩ F)<br /> </p> </div> </div> </div> -<div id="outline-container-org44b1b96" class="outline-3"> -<h3 id="org44b1b96">Proprieties :</h3> -<div class="outline-text-3" id="text-org44b1b96"> +<div id="outline-container-org691c863" class="outline-3"> +<h3 id="org691c863">Proprieties :</h3> +<div class="outline-text-3" id="text-org691c863"> <p> Let E,F and G be 3 sets. We have :<br /> </p> </div> -<div id="outline-container-orga3eac79" class="outline-4"> -<h4 id="orga3eac79">Commutativity:</h4> -<div class="outline-text-4" id="text-orga3eac79"> +<div id="outline-container-org9cc9f31" class="outline-4"> +<h4 id="org9cc9f31">Commutativity:</h4> +<div class="outline-text-4" id="text-org9cc9f31"> <p> E ∩ F = F ∩ E<br /> E ∪ F = F ∪ E<br /> </p> </div> </div> -<div id="outline-container-org1a9121a" class="outline-4"> -<h4 id="org1a9121a">Associativity:</h4> -<div class="outline-text-4" id="text-org1a9121a"> +<div id="outline-container-org471083b" class="outline-4"> +<h4 id="org471083b">Associativity:</h4> +<div class="outline-text-4" id="text-org471083b"> <p> E ∩ (F ∩ G) = (E ∩ F) ∩ G<br /> E ∪ (F ∪ G) = (E ∪ F) ∪ G<br /> </p> </div> </div> -<div id="outline-container-orgaf7fe3b" class="outline-4"> -<h4 id="orgaf7fe3b">Distributivity:</h4> -<div class="outline-text-4" id="text-orgaf7fe3b"> +<div id="outline-container-orge63be10" class="outline-4"> +<h4 id="orge63be10">Distributivity:</h4> +<div class="outline-text-4" id="text-orge63be10"> <p> E ∩ (F ∪ G) = (E ∩ F) ∪ (E ∩ G)<br /> E ∪ (F ∩ G) = (E ∪ F) ∩ (E ∪ G)<br /> </p> </div> </div> -<div id="outline-container-org658b728" class="outline-4"> -<h4 id="org658b728">Lois de Morgan:</h4> -<div class="outline-text-4" id="text-org658b728"> +<div id="outline-container-orgfb01947" class="outline-4"> +<h4 id="orgfb01947">Lois de Morgan:</h4> +<div class="outline-text-4" id="text-orgfb01947"> <p> If E ⊂ G and F ⊂ G ;<br /> </p> @@ -1390,33 +1391,33 @@ If E ⊂ G and F ⊂ G ;<br /> </p> </div> </div> -<div id="outline-container-org4f0dd58" class="outline-4"> -<h4 id="org4f0dd58">An other one:</h4> -<div class="outline-text-4" id="text-org4f0dd58"> +<div id="outline-container-orge1a41eb" class="outline-4"> +<h4 id="orge1a41eb">An other one:</h4> +<div class="outline-text-4" id="text-orge1a41eb"> <p> E - (F ∩ G) = (E-F) ∪ (E-G) ; E - (F ∪ G) = (E-F) ∩ (E-G)<br /> </p> </div> </div> -<div id="outline-container-orgdd60033" class="outline-4"> -<h4 id="orgdd60033">An other one:</h4> -<div class="outline-text-4" id="text-orgdd60033"> +<div id="outline-container-org9939b0f" class="outline-4"> +<h4 id="org9939b0f">An other one:</h4> +<div class="outline-text-4" id="text-org9939b0f"> <p> E ∩ ∅ = ∅ ; E ∪ ∅ = E<br /> </p> </div> </div> -<div id="outline-container-orgbf5feb1" class="outline-4"> -<h4 id="orgbf5feb1">And an other one:</h4> -<div class="outline-text-4" id="text-orgbf5feb1"> +<div id="outline-container-org90bfdc4" class="outline-4"> +<h4 id="org90bfdc4">And an other one:</h4> +<div class="outline-text-4" id="text-org90bfdc4"> <p> E ∩ (F Δ G) = (E ∩ F) Δ (E ∩ G)<br /> </p> </div> </div> -<div id="outline-container-orgefabd47" class="outline-4"> -<h4 id="orgefabd47">And the last one:</h4> -<div class="outline-text-4" id="text-orgefabd47"> +<div id="outline-container-org1e49001" class="outline-4"> +<h4 id="org1e49001">And the last one:</h4> +<div class="outline-text-4" id="text-org1e49001"> <p> E Δ ∅ = E ; E Δ E = ∅<br /> </p> @@ -1424,16 +1425,16 @@ E Δ ∅ = E ; E Δ E = ∅<br /> </div> </div> </div> -<div id="outline-container-orgc9ed06c" class="outline-2"> -<h2 id="orgc9ed06c">5eme cours: L’ensemble des parties d’un ensemble <i>Oct 16</i></h2> -<div class="outline-text-2" id="text-orgc9ed06c"> +<div id="outline-container-org272eca3" class="outline-2"> +<h2 id="org272eca3">5eme cours: L’ensemble des parties d’un ensemble <i>Oct 16</i></h2> +<div class="outline-text-2" id="text-org272eca3"> <p> Let E be a set. We define P(E) as the set of all parts of E : <b>P(E) = {X/X ⊂ E}</b><br /> </p> </div> -<div id="outline-container-org7a1da3c" class="outline-4"> -<h4 id="org7a1da3c">Notes :</h4> -<div class="outline-text-4" id="text-org7a1da3c"> +<div id="outline-container-org5b29e32" class="outline-4"> +<h4 id="org5b29e32">Notes :</h4> +<div class="outline-text-4" id="text-org5b29e32"> <p> ∅ ∈ P(E) ; E ∈ P(E)<br /> </p> @@ -1444,17 +1445,17 @@ cardinal E = n <i>The number of terms in E</i> , cardinal P(E) = 2^n <i>The numb </p> </div> </div> -<div id="outline-container-orga6e5f8a" class="outline-4"> -<h4 id="orga6e5f8a">Examples :</h4> -<div class="outline-text-4" id="text-orga6e5f8a"> +<div id="outline-container-org5636bd8" class="outline-4"> +<h4 id="org5636bd8">Examples :</h4> +<div class="outline-text-4" id="text-org5636bd8"> <p> E = {a,b,c} ; P(E)={∅, {a}, {b}, {c}, {a,b}, {b,c}, {a,c}, {a,b,c}}<br /> </p> </div> </div> -<div id="outline-container-org7286ec5" class="outline-3"> -<h3 id="org7286ec5">Partition of a set :</h3> -<div class="outline-text-3" id="text-org7286ec5"> +<div id="outline-container-orgd8cb2c3" class="outline-3"> +<h3 id="orgd8cb2c3">Partition of a set :</h3> +<div class="outline-text-3" id="text-orgd8cb2c3"> <p> We say that <b>A</b> is a partition of E if:<br /> </p> @@ -1465,16 +1466,16 @@ We say that <b>A</b> is a partition of E if:<br /> </ol> </div> </div> -<div id="outline-container-orgd8e00ac" class="outline-3"> -<h3 id="orgd8e00ac">Cartesian products :</h3> -<div class="outline-text-3" id="text-orgd8e00ac"> +<div id="outline-container-orgf40404d" class="outline-3"> +<h3 id="orgf40404d">Cartesian products :</h3> +<div class="outline-text-3" id="text-orgf40404d"> <p> Let E and F be two sets, the set EXF = {(x,y)/ x ∈ E AND y ∈ F} is called the Cartesian product of E and F<br /> </p> </div> -<div id="outline-container-orgdb491ee" class="outline-4"> -<h4 id="orgdb491ee">Example :</h4> -<div class="outline-text-4" id="text-orgdb491ee"> +<div id="outline-container-orgd526cb8" class="outline-4"> +<h4 id="orgd526cb8">Example :</h4> +<div class="outline-text-4" id="text-orgd526cb8"> <p> A = {4,5} ; B= {4,5,6} ; AxB = {(4,4), (4,5), (4,6), (5,4), (5,5), (5,6)}<br /> </p> @@ -1485,9 +1486,9 @@ BxA = {(4,4), (4,5), (5,4), (5,5), (6,4), (6,5)} ; Therefore AxB ≠ BxA<br /> </p> </div> </div> -<div id="outline-container-org28c23b2" class="outline-4"> -<h4 id="org28c23b2">Some proprieties:</h4> -<div class="outline-text-4" id="text-org28c23b2"> +<div id="outline-container-org56dd088" class="outline-4"> +<h4 id="org56dd088">Some proprieties:</h4> +<div class="outline-text-4" id="text-org56dd088"> <ol class="org-ol"> <li>ExF = ∅ ⇔ E=∅ OR F=∅<br /></li> <li>ExF = FxE ⇔ E=F OR E=∅ OR F=∅<br /></li> @@ -1500,21 +1501,21 @@ BxA = {(4,4), (4,5), (5,4), (5,5), (6,4), (6,5)} ; Therefore AxB ≠ BxA<br /> </div> </div> </div> -<div id="outline-container-orgfdbe4f3" class="outline-2"> -<h2 id="orgfdbe4f3">Binary relations in a set :</h2> -<div class="outline-text-2" id="text-orgfdbe4f3"> +<div id="outline-container-org5ee4278" class="outline-2"> +<h2 id="org5ee4278">Binary relations in a set :</h2> +<div class="outline-text-2" id="text-org5ee4278"> </div> -<div id="outline-container-org3696656" class="outline-3"> -<h3 id="org3696656">Definition :</h3> -<div class="outline-text-3" id="text-org3696656"> +<div id="outline-container-orgddc9af6" class="outline-3"> +<h3 id="orgddc9af6">Definition :</h3> +<div class="outline-text-3" id="text-orgddc9af6"> <p> Let E be a set and x,y ∈ E. If there exists a link between x and y, we say that they are tied by a relation <b>R</b> and we write <b>xRy</b><br /> </p> </div> </div> -<div id="outline-container-orgd32f673" class="outline-3"> -<h3 id="orgd32f673">Proprieties :</h3> -<div class="outline-text-3" id="text-orgd32f673"> +<div id="outline-container-orge65424e" class="outline-3"> +<h3 id="orge65424e">Proprieties :</h3> +<div class="outline-text-3" id="text-orge65424e"> <p> Let E be a set and R a relation defined in E<br /> </p> @@ -1526,16 +1527,16 @@ Let E be a set and R a relation defined in E<br /> </ol> </div> </div> -<div id="outline-container-org22f460a" class="outline-3"> -<h3 id="org22f460a">Equivalence relationship :</h3> -<div class="outline-text-3" id="text-org22f460a"> +<div id="outline-container-orgd7877d3" class="outline-3"> +<h3 id="orgd7877d3">Equivalence relationship :</h3> +<div class="outline-text-3" id="text-orgd7877d3"> <p> We say that R is a relation of equivalence in E if its reflexive, symetrical and transitive<br /> </p> </div> -<div id="outline-container-org68ddde2" class="outline-4"> -<h4 id="org68ddde2">Equivalence class :</h4> -<div class="outline-text-4" id="text-org68ddde2"> +<div id="outline-container-org85cf025" class="outline-4"> +<h4 id="org85cf025">Equivalence class :</h4> +<div class="outline-text-4" id="text-org85cf025"> <p> Let R be a relation of equivalence in E and a ∈ E, we call equivalence class of <b>a</b>, and we write ̅a or ȧ, or cl a the following set :<br /> </p> @@ -1546,8 +1547,8 @@ Let R be a relation of equivalence in E and a ∈ E, we call equivalence class o </p> </div> <ul class="org-ul"> -<li><a id="org431774d"></a>The quotient set :<br /> -<div class="outline-text-5" id="text-org431774d"> +<li><a id="orga316a01"></a>The quotient set :<br /> +<div class="outline-text-5" id="text-orga316a01"> <p> E/R = {̅a , a ∈ E}<br /> </p> @@ -1556,9 +1557,9 @@ E/R = {̅a , a ∈ E}<br /> </ul> </div> </div> -<div id="outline-container-orge976c7e" class="outline-3"> -<h3 id="orge976c7e">Order relationship :</h3> -<div class="outline-text-3" id="text-orge976c7e"> +<div id="outline-container-orge18dcc7" class="outline-3"> +<h3 id="orge18dcc7">Order relationship :</h3> +<div class="outline-text-3" id="text-orge18dcc7"> <p> Let E be a set and R be a relation defined in E. We say that R is a relation of order if its reflexive, anti-symetrical and transitive.<br /> </p> @@ -1567,9 +1568,9 @@ Let E be a set and R be a relation defined in E. We say that R is a relation of <li>The order R is called partial if ∃ x,y ∈ E xR̅y AND yR̅x<br /></li> </ol> </div> -<div id="outline-container-org1f19847" class="outline-4"> -<h4 id="org1f19847"><span class="todo TODO">TODO</span> Examples :</h4> -<div class="outline-text-4" id="text-org1f19847"> +<div id="outline-container-org60d471a" class="outline-4"> +<h4 id="org60d471a"><span class="todo TODO">TODO</span> Examples :</h4> +<div class="outline-text-4" id="text-org60d471a"> <p> ∀x,y ∈ ℝ , xRy ⇔ x²-y²=x-y<br /> </p> @@ -1581,17 +1582,17 @@ Let E be a set and R be a relation defined in E. We say that R is a relation of </div> </div> </div> -<div id="outline-container-orgc956a73" class="outline-2"> -<h2 id="orgc956a73">TP exercices <i>Oct 20</i> :</h2> -<div class="outline-text-2" id="text-orgc956a73"> +<div id="outline-container-org77de6e3" class="outline-2"> +<h2 id="org77de6e3">TP exercices <i>Oct 20</i> :</h2> +<div class="outline-text-2" id="text-org77de6e3"> </div> -<div id="outline-container-org15b0e75" class="outline-3"> -<h3 id="org15b0e75">Exercice 3 :</h3> -<div class="outline-text-3" id="text-org15b0e75"> +<div id="outline-container-org3ca8006" class="outline-3"> +<h3 id="org3ca8006">Exercice 3 :</h3> +<div class="outline-text-3" id="text-org3ca8006"> </div> -<div id="outline-container-orgb132892" class="outline-4"> -<h4 id="orgb132892">Question 3</h4> -<div class="outline-text-4" id="text-orgb132892"> +<div id="outline-container-orgad95ec3" class="outline-4"> +<h4 id="orgad95ec3">Question 3</h4> +<div class="outline-text-4" id="text-orgad95ec3"> <p> Montrer par l’absurde que P : ∀x ∈ ℝ*, √(4+x³) ≠ 2 + x³/4 est vraies<br /> </p> @@ -1608,13 +1609,13 @@ x = 0 . Or, x appartiens a ℝ\{0}, donc P̅ est fausse. Ce qui est equivalent a </div> </div> </div> -<div id="outline-container-org9a4006b" class="outline-3"> -<h3 id="org9a4006b">Exercice 4 :</h3> -<div class="outline-text-3" id="text-org9a4006b"> +<div id="outline-container-org8180ae0" class="outline-3"> +<h3 id="org8180ae0">Exercice 4 :</h3> +<div class="outline-text-3" id="text-org8180ae0"> </div> -<div id="outline-container-org43cf6d6" class="outline-4"> -<h4 id="org43cf6d6"><span class="done DONE">DONE</span> Question 1 :</h4> -<div class="outline-text-4" id="text-org43cf6d6"> +<div id="outline-container-orgfe0b1e2" class="outline-4"> +<h4 id="orgfe0b1e2"><span class="done DONE">DONE</span> Question 1 :</h4> +<div class="outline-text-4" id="text-orgfe0b1e2"> <p class="verse"> ∀ n ∈ ℕ* , (n ,k=1)Σ1/k(k+1) = 1 - 1/1+n<br /> P(n) : (n ,k=1)Σ1/k(k+1) = 1 - 1/1+n<br /> @@ -1642,17 +1643,17 @@ De (a) et (b) on conclus que la proposition de départ est vraie<br /> </div> </div> </div> -<div id="outline-container-org429ab91" class="outline-2"> -<h2 id="org429ab91">Chapter 3 : Applications</h2> -<div class="outline-text-2" id="text-org429ab91"> +<div id="outline-container-org2d6e0ba" class="outline-2"> +<h2 id="org2d6e0ba">Chapter 3 : Applications</h2> +<div class="outline-text-2" id="text-org2d6e0ba"> </div> -<div id="outline-container-org4a4b3cf" class="outline-3"> -<h3 id="org4a4b3cf">3.1 Generalities about applications :</h3> -<div class="outline-text-3" id="text-org4a4b3cf"> +<div id="outline-container-orga5be12f" class="outline-3"> +<h3 id="orga5be12f">3.1 Generalities about applications :</h3> +<div class="outline-text-3" id="text-orga5be12f"> </div> -<div id="outline-container-org0cac0c6" class="outline-4"> -<h4 id="org0cac0c6">Definition :</h4> -<div class="outline-text-4" id="text-org0cac0c6"> +<div id="outline-container-org805d7bc" class="outline-4"> +<h4 id="org805d7bc">Definition :</h4> +<div class="outline-text-4" id="text-org805d7bc"> <p> Let E and F be two sets.<br /> </p> @@ -1674,10 +1675,10 @@ f : E —> F<br /> </ol> </div> <ul class="org-ul"> -<li><a id="org1570343"></a>Some examples :<br /> +<li><a id="org2936c19"></a>Some examples :<br /> <ul class="org-ul"> -<li><a id="orgdafc0e8"></a>Ex1:<br /> -<div class="outline-text-6" id="text-orgdafc0e8"> +<li><a id="orgd77c836"></a>Ex1:<br /> +<div class="outline-text-6" id="text-orgd77c836"> <p class="verse"> f : ℝ —> ℝ<br />     x —> f(x) = (x-1)/x<br /> @@ -1685,8 +1686,8 @@ is a function, because 0 does NOT have a corresponding element using that relati </p> </div> </li> -<li><a id="orgcc5e730"></a>Ex2:<br /> -<div class="outline-text-6" id="text-orgcc5e730"> +<li><a id="orga45fd32"></a>Ex2:<br /> +<div class="outline-text-6" id="text-orga45fd32"> <p class="verse"> f : ℝ<sup>*</sup> —> ℝ<br />     x —> f(x)= (x-1)/x<br /> @@ -1698,9 +1699,9 @@ is, however, an application<br /> </li> </ul> </div> -<div id="outline-container-orgc048e93" class="outline-4"> -<h4 id="orgc048e93">Restriction and prolongation of an application :</h4> -<div class="outline-text-4" id="text-orgc048e93"> +<div id="outline-container-org7947331" class="outline-4"> +<h4 id="org7947331">Restriction and prolongation of an application :</h4> +<div class="outline-text-4" id="text-org7947331"> <p> Let f : E -> F an application and E<sub>1</sub> ⊂ E therefore :<br /> </p> @@ -1712,8 +1713,8 @@ g is called the <b>restriction</b> of f to E<sub>1</sub>. And f is called the <b </p> </div> <ul class="org-ul"> -<li><a id="org771da73"></a>Example<br /> -<div class="outline-text-5" id="text-org771da73"> +<li><a id="org7c848c6"></a>Example<br /> +<div class="outline-text-5" id="text-org7c848c6"> <p class="verse"> f : ℝ —> ℝ<br />     x —> f(x) = x<sup>2</sup><br /> @@ -1727,9 +1728,9 @@ g is called the <b>restriction</b> of f to ℝ^{</del>}. And f is called the <b> </li> </ul> </div> -<div id="outline-container-org8e61361" class="outline-4"> -<h4 id="org8e61361">Composition of applications :</h4> -<div class="outline-text-4" id="text-org8e61361"> +<div id="outline-container-orgd94bc69" class="outline-4"> +<h4 id="orgd94bc69">Composition of applications :</h4> +<div class="outline-text-4" id="text-orgd94bc69"> <p> Let E,F, and G be three sets, f: E -> F and g: F -> G are two applications. We define their composition, symbolized by g<sub>o</sub>f as follow :<br /> </p> @@ -1741,9 +1742,9 @@ g<sub>o</sub>f : E -> G . ∀x ∈ E (g<sub>o</sub>f)<sub>(x)</sub>= g(f(x))< </div> </div> </div> -<div id="outline-container-org5c096db" class="outline-3"> -<h3 id="org5c096db">3.2 Injection, surjection and bijection :</h3> -<div class="outline-text-3" id="text-org5c096db"> +<div id="outline-container-org257d05a" class="outline-3"> +<h3 id="org257d05a">3.2 Injection, surjection and bijection :</h3> +<div class="outline-text-3" id="text-org257d05a"> <p> Let f: E -> F be an application :<br /> </p> @@ -1753,9 +1754,9 @@ Let f: E -> F be an application :<br /> <li>We say that if is bijective if it’s both injective and surjective at the same time.<br /></li> </ol> </div> -<div id="outline-container-org4162b56" class="outline-4"> -<h4 id="org4162b56">Proposition :</h4> -<div class="outline-text-4" id="text-org4162b56"> +<div id="outline-container-org1612e09" class="outline-4"> +<h4 id="org1612e09">Proposition :</h4> +<div class="outline-text-4" id="text-org1612e09"> <p> Let f : E -> F be an application. Therefore:<br /> </p> @@ -1767,21 +1768,21 @@ Let f : E -> F be an application. Therefore:<br /> </div> </div> </div> -<div id="outline-container-org736de6c" class="outline-3"> -<h3 id="org736de6c">3.3 Reciprocal applications :</h3> -<div class="outline-text-3" id="text-org736de6c"> +<div id="outline-container-orgebdf518" class="outline-3"> +<h3 id="orgebdf518">3.3 Reciprocal applications :</h3> +<div class="outline-text-3" id="text-orgebdf518"> </div> -<div id="outline-container-orgafb7f85" class="outline-4"> -<h4 id="orgafb7f85">Def :</h4> -<div class="outline-text-4" id="text-orgafb7f85"> +<div id="outline-container-orgf072e42" class="outline-4"> +<h4 id="orgf072e42">Def :</h4> +<div class="outline-text-4" id="text-orgf072e42"> <p> Let f : E -> F a bijective application. So there exists an application named f<sup>-1</sup> : F -> E such as : y = f(x) ⇔ x = f<sup>-1</sup>(y)<br /> </p> </div> </div> -<div id="outline-container-orgec3a3d6" class="outline-4"> -<h4 id="orgec3a3d6">Theorem :</h4> -<div class="outline-text-4" id="text-orgec3a3d6"> +<div id="outline-container-org244b352" class="outline-4"> +<h4 id="org244b352">Theorem :</h4> +<div class="outline-text-4" id="text-org244b352"> <p> Let f : E -> F be a bijective application. Therefore its reciprocal f<sup>-1</sup> verifies : f<sup>-1</sup><sub>o</sub>f=Id<sub>E </sub>; f<sub>o</sub>f<sup>-1</sup>=Id<sub>F</sub> Or :<br /> </p> @@ -1792,9 +1793,9 @@ Id<sub>E</sub> : E -> E ; x -> Id<sub>E</sub>(x) = x<br /> </p> </div> </div> -<div id="outline-container-org940e2d6" class="outline-4"> -<h4 id="org940e2d6">Some proprieties :</h4> -<div class="outline-text-4" id="text-org940e2d6"> +<div id="outline-container-org1479c0e" class="outline-4"> +<h4 id="org1479c0e">Some proprieties :</h4> +<div class="outline-text-4" id="text-org1479c0e"> <ol class="org-ol"> <li>(f<sup>-1</sup>)<sup>-1</sup> = f<br /></li> <li>(g<sub>o</sub>f)⁻¹ = f⁻¹<sub>o</sub>g⁻¹<br /></li> @@ -1803,13 +1804,13 @@ Id<sub>E</sub> : E -> E ; x -> Id<sub>E</sub>(x) = x<br /> </div> </div> </div> -<div id="outline-container-org2d173c2" class="outline-3"> -<h3 id="org2d173c2">3.4 Direct Image and reciprocal Image :</h3> -<div class="outline-text-3" id="text-org2d173c2"> +<div id="outline-container-orgaf81bb3" class="outline-3"> +<h3 id="orgaf81bb3">3.4 Direct Image and reciprocal Image :</h3> +<div class="outline-text-3" id="text-orgaf81bb3"> </div> -<div id="outline-container-org769c809" class="outline-4"> -<h4 id="org769c809">Direct Image :</h4> -<div class="outline-text-4" id="text-org769c809"> +<div id="outline-container-org87b91e2" class="outline-4"> +<h4 id="org87b91e2">Direct Image :</h4> +<div class="outline-text-4" id="text-org87b91e2"> <p> Let f: E-> F be an application and A ⊂ E. We call a direct image of A by f, and we symbolize as f(A) the subset of F defined by :<br /> </p> @@ -1820,8 +1821,8 @@ f(A) = {f(x)/ x ∈ A} ; = { y ∈ F ∃ x ∈ A y=f(x)}<br /> </p> </div> <ul class="org-ul"> -<li><a id="org492a9ac"></a>Example :<br /> -<div class="outline-text-5" id="text-org492a9ac"> +<li><a id="orgb5bc08c"></a>Example :<br /> +<div class="outline-text-5" id="text-orgb5bc08c"> <p class="verse"> f: ℝ -> ℝ<br />    x -> f(x) = x²<br /> @@ -1832,9 +1833,9 @@ f(A) = {f(0), f(4)} = {0, 16}<br /> </li> </ul> </div> -<div id="outline-container-org7d705d3" class="outline-4"> -<h4 id="org7d705d3">Reciprocal image :</h4> -<div class="outline-text-4" id="text-org7d705d3"> +<div id="outline-container-org500bc40" class="outline-4"> +<h4 id="org500bc40">Reciprocal image :</h4> +<div class="outline-text-4" id="text-org500bc40"> <p> Let f: E -> F be an application and B ⊂ F. We call the reciprocal image of E by F the subset f<sup>-1</sup>(B) :<br /> </p> @@ -1845,8 +1846,8 @@ f<sup>-1</sup>(B) = {x ∈ E/f(x) ∈ B} ; x ∈ f<sup>-1</sup>(B) ⇔ f(x) ∈ </p> </div> <ul class="org-ul"> -<li><a id="org686c1dc"></a>Example :<br /> -<div class="outline-text-5" id="text-org686c1dc"> +<li><a id="org885d21d"></a>Example :<br /> +<div class="outline-text-5" id="text-org885d21d"> <p class="verse"> f: ℝ -> ℝ<br />    x -> f(x) = x²<br /> @@ -1863,7 +1864,7 @@ f<sup>-1</sup>(B) = {1,-1,2,-2,3,-3}<br /> </div> <div id="postamble" class="status"> <p class="author">Author: Crystal</p> -<p class="date">Created: 2023-10-23 Mon 19:39</p> +<p class="date">Created: 2023-11-01 Wed 20:17</p> </div> </body> </html> \ No newline at end of file diff --git a/uni_notes/alsd.html b/uni_notes/alsd.html index 128e304..804d386 100755 --- a/uni_notes/alsd.html +++ b/uni_notes/alsd.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <head> -<!-- 2023-11-01 Wed 20:10 --> +<!-- 2023-11-01 Wed 20:17 --> <meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <title>ALSD1</title> @@ -11,7 +11,7 @@ <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"/> -<link rel="icon" type="image/x-icon" href="https://crystal.tilde.institute/favicon.ico"> +<link rel="icon" type="image/x-icon" href="https://crystal.tilde.institute/favicon.png"> </head> <body> <div id="org-div-home-and-up"> @@ -24,20 +24,20 @@ <h2>Table of Contents</h2> <div id="text-table-of-contents" role="doc-toc"> <ul> -<li><a href="#org397516f">Contenu de la Matiére</a> +<li><a href="#org99d5220">Contenu de la Matiére</a> <ul> -<li><a href="#orgabf6101">Chapitre 1: Elements de Base</a></li> -<li><a href="#org91ff844">Chapitre 2: Présentation du formalisme Algorithmique</a></li> -<li><a href="#org21033bb">Chapitre 3: Eléments de base du language C</a></li> -<li><a href="#org18ac720">Chapitre 4: Modularité( Fonction et Procédure )</a></li> -<li><a href="#orgfd737ff">Chapitre 5: Les structures des données statiques</a></li> +<li><a href="#orgbd002d7">Chapitre 1: Elements de Base</a></li> +<li><a href="#orgeb962fb">Chapitre 2: Présentation du formalisme Algorithmique</a></li> +<li><a href="#org7a1fb51">Chapitre 3: Eléments de base du language C</a></li> +<li><a href="#org5f789aa">Chapitre 4: Modularité( Fonction et Procédure )</a></li> +<li><a href="#org330830d">Chapitre 5: Les structures des données statiques</a></li> </ul> </li> -<li><a href="#org18cb30a">Premier cours : Algorithmes <i>Oct 1</i> :</a> +<li><a href="#orgced3d78">Premier cours : Algorithmes <i>Oct 1</i> :</a> <ul> -<li><a href="#orga89bea0">Définition d’un algorithm :</a> +<li><a href="#orga9ff303">Définition d’un algorithm :</a> <ul> -<li><a href="#orge424ff9">Example d’un Algo : Résolution d’une équation du second ordre (ax²+bx+c=0)</a></li> +<li><a href="#org38de9fd">Example d’un Algo : Résolution d’une équation du second ordre (ax²+bx+c=0)</a></li> </ul> </li> </ul> @@ -45,13 +45,13 @@ </ul> </div> </div> -<div id="outline-container-org397516f" class="outline-2"> -<h2 id="org397516f">Contenu de la Matiére</h2> -<div class="outline-text-2" id="text-org397516f"> +<div id="outline-container-org99d5220" class="outline-2"> +<h2 id="org99d5220">Contenu de la Matiére</h2> +<div class="outline-text-2" id="text-org99d5220"> </div> -<div id="outline-container-orgabf6101" class="outline-3"> -<h3 id="orgabf6101">Chapitre 1: Elements de Base</h3> -<div class="outline-text-3" id="text-orgabf6101"> +<div id="outline-container-orgbd002d7" class="outline-3"> +<h3 id="orgbd002d7">Chapitre 1: Elements de Base</h3> +<div class="outline-text-3" id="text-orgbd002d7"> <ul class="org-ul"> <li>Algorithmique, procésseur, action.<br /></li> <li>Programme et languages de programmation.<br /></li> @@ -59,33 +59,33 @@ </ul> </div> </div> -<div id="outline-container-org91ff844" class="outline-3"> -<h3 id="org91ff844">Chapitre 2: Présentation du formalisme Algorithmique</h3> +<div id="outline-container-orgeb962fb" class="outline-3"> +<h3 id="orgeb962fb">Chapitre 2: Présentation du formalisme Algorithmique</h3> </div> -<div id="outline-container-org21033bb" class="outline-3"> -<h3 id="org21033bb">Chapitre 3: Eléments de base du language C</h3> +<div id="outline-container-org7a1fb51" class="outline-3"> +<h3 id="org7a1fb51">Chapitre 3: Eléments de base du language C</h3> </div> -<div id="outline-container-org18ac720" class="outline-3"> -<h3 id="org18ac720">Chapitre 4: Modularité( Fonction et Procédure )</h3> +<div id="outline-container-org5f789aa" class="outline-3"> +<h3 id="org5f789aa">Chapitre 4: Modularité( Fonction et Procédure )</h3> </div> -<div id="outline-container-orgfd737ff" class="outline-3"> -<h3 id="orgfd737ff">Chapitre 5: Les structures des données statiques</h3> +<div id="outline-container-org330830d" class="outline-3"> +<h3 id="org330830d">Chapitre 5: Les structures des données statiques</h3> </div> </div> -<div id="outline-container-org18cb30a" class="outline-2"> -<h2 id="org18cb30a">Premier cours : Algorithmes <i>Oct 1</i> :</h2> -<div class="outline-text-2" id="text-org18cb30a"> +<div id="outline-container-orgced3d78" class="outline-2"> +<h2 id="orgced3d78">Premier cours : Algorithmes <i>Oct 1</i> :</h2> +<div class="outline-text-2" id="text-orgced3d78"> </div> -<div id="outline-container-orga89bea0" class="outline-3"> -<h3 id="orga89bea0">Définition d’un algorithm :</h3> -<div class="outline-text-3" id="text-orga89bea0"> +<div id="outline-container-orga9ff303" class="outline-3"> +<h3 id="orga9ff303">Définition d’un algorithm :</h3> +<div class="outline-text-3" id="text-orga9ff303"> <p> Un ensemble d’opérations ecrites dans le language naturel.<br /> </p> </div> -<div id="outline-container-orge424ff9" class="outline-4"> -<h4 id="orge424ff9">Example d’un Algo : Résolution d’une équation du second ordre (ax²+bx+c=0)</h4> -<div class="outline-text-4" id="text-orge424ff9"> +<div id="outline-container-org38de9fd" class="outline-4"> +<h4 id="org38de9fd">Example d’un Algo : Résolution d’une équation du second ordre (ax²+bx+c=0)</h4> +<div class="outline-text-4" id="text-org38de9fd"> <ol class="org-ol"> <li>Si a=0 ET b=0 alors <b>l’équation n’est pas du 2nd ordre</b>.<br /></li> <li>Si a=0 et b≠0 alors <b>x= -c/5</b> .<br /></li> @@ -103,7 +103,7 @@ Un ensemble d’opérations ecrites dans le language naturel.<br /> </div> <div id="postamble" class="status"> <p class="author">Author: Crystal</p> -<p class="date">Created: 2023-11-01 Wed 20:10</p> +<p class="date">Created: 2023-11-01 Wed 20:17</p> </div> </body> </html> \ No newline at end of file diff --git a/uni_notes/analyse.html b/uni_notes/analyse.html index 1986e02..c0f1f20 100755 --- a/uni_notes/analyse.html +++ b/uni_notes/analyse.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <head> -<!-- 2023-11-01 Wed 20:10 --> +<!-- 2023-11-01 Wed 20:16 --> <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> @@ -11,7 +11,7 @@ <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"/> -<link rel="icon" type="image/x-icon" href="https://crystal.tilde.institute/favicon.ico"> +<link rel="icon" type="image/x-icon" href="https://crystal.tilde.institute/favicon.png"> </head> <body> <div id="org-div-home-and-up"> @@ -24,176 +24,176 @@ <h2>Table of Contents</h2> <div id="text-table-of-contents" role="doc-toc"> <ul> -<li><a href="#org69b61eb">Contenu de la Matiére</a> +<li><a href="#org96a1915">Contenu de la Matiére</a> <ul> -<li><a href="#org0a8600d">Chapitre 1 : Quelque propriétés de ℝ</a></li> -<li><a href="#org85fd1b6">Chapitre 2 : Les suites numériques réelles</a></li> -<li><a href="#org0f00d84">Chapitre 3 : Limites et continuité des fonctions réelles d’une variable réelle</a></li> -<li><a href="#org65466d0">Chapitre 4 : La dérivabilité et son interprétation géometrique</a></li> -<li><a href="#org12481b8">Chapitre 5 : Les fonctions trigonométriques réciproques, fonctions hypérboliques réciproques</a></li> +<li><a href="#org92a5c1e">Chapitre 1 : Quelque propriétés de ℝ</a></li> +<li><a href="#org43abd6f">Chapitre 2 : Les suites numériques réelles</a></li> +<li><a href="#orgb7dbd4d">Chapitre 3 : Limites et continuité des fonctions réelles d’une variable réelle</a></li> +<li><a href="#orgb39022d">Chapitre 4 : La dérivabilité et son interprétation géometrique</a></li> +<li><a href="#orgbfa8dc6">Chapitre 5 : Les fonctions trigonométriques réciproques, fonctions hypérboliques réciproques</a></li> </ul> </li> -<li><a href="#org3fc2976">Premier cours : Quelque propriétés de ℝ <i>Sep 26</i> :</a> +<li><a href="#org0e8210e">Premier cours : Quelque propriétés de ℝ <i>Sep 26</i> :</a> <ul> -<li><a href="#org4e71cb9">La loi de composition interne dans E :</a> +<li><a href="#org4b5ef0e">La loi de composition interne dans E :</a> <ul> -<li><a href="#orgbc38c17"><b>Example : Addition</b></a></li> -<li><a href="#org32a22fa"><b>Example : soustraction</b></a></li> +<li><a href="#org26ad93c"><b>Example : Addition</b></a></li> +<li><a href="#org716f99d"><b>Example : soustraction</b></a></li> </ul> </li> -<li><a href="#org19f1c04">La loi de composition externe dans E :</a></li> -<li><a href="#org24ed87c">Groupes :</a> +<li><a href="#orgcd239ec">La loi de composition externe dans E :</a></li> +<li><a href="#org6aa8256">Groupes :</a> <ul> -<li><a href="#org8f69286">Il contiens un élement neutre</a></li> -<li><a href="#org6747a03">Il contiens un élément symétrique</a></li> -<li><a href="#org4bf37c2">@ est cummutative :</a></li> +<li><a href="#org3a88117">Il contiens un élement neutre</a></li> +<li><a href="#orgf17cd87">Il contiens un élément symétrique</a></li> +<li><a href="#org541a8aa">@ est cummutative :</a></li> </ul> </li> -<li><a href="#orgd89b7dc">Anneaux :</a> +<li><a href="#org325ac76">Anneaux :</a> <ul> -<li><a href="#org0974b29">(E ; @) est un groupe cummutatif</a></li> -<li><a href="#org9b00b5e">! est une loi associative :</a></li> -<li><a href="#org248e491">Distribution de ! par rapport à @ :</a></li> -<li><a href="#org6cbe38c">L’existance d’un élèment neutre de ! :</a></li> -<li><a href="#orgd1b5bee">! est cummutative :</a></li> +<li><a href="#orgb12d61b">(E ; @) est un groupe cummutatif</a></li> +<li><a href="#org2f6c910">! est une loi associative :</a></li> +<li><a href="#org288714a">Distribution de ! par rapport à @ :</a></li> +<li><a href="#org92b8438">L’existance d’un élèment neutre de ! :</a></li> +<li><a href="#org12cac33">! est cummutative :</a></li> </ul> </li> -<li><a href="#org7fdacd4">Corps :</a> +<li><a href="#orgbdac47c">Corps :</a> <ul> -<li><a href="#org3853ce7">La symétrie :</a></li> +<li><a href="#orgea3147a">La symétrie :</a></li> </ul> </li> -<li><a href="#org3922e65">Exercice : (ℝ, +, x) corps ou pas ?</a> +<li><a href="#org012a6fe">Exercice : (ℝ, +, x) corps ou pas ?</a> <ul> -<li><a href="#org13f7b7e">Est-ce un Anneau ?</a></li> -<li><a href="#org90e2795">Est-ce un corps ?</a></li> +<li><a href="#org0934303">Est-ce un Anneau ?</a></li> +<li><a href="#orgb7020c8">Est-ce un corps ?</a></li> </ul> </li> </ul> </li> -<li><a href="#org76e7ef3">2nd cours :L’ordre dans ℝ, Majorant, minorant, borne superieure, borne inférieure <i>Oct 3</i> :</a> +<li><a href="#orgabdfea2">2nd cours :L’ordre dans ℝ, Majorant, minorant, borne superieure, borne inférieure <i>Oct 3</i> :</a> <ul> -<li><a href="#org94449fa">L’ordre dans ℝ</a> +<li><a href="#org30b59ae">L’ordre dans ℝ</a> <ul> -<li><a href="#org17e010d">Exemples :</a></li> +<li><a href="#org86a0035">Exemples :</a></li> </ul> </li> -<li><a href="#orga26a211">Majorant, minorant, borne supérieure, borne inférieure</a> +<li><a href="#org43ad665">Majorant, minorant, borne supérieure, borne inférieure</a> <ul> -<li><a href="#org3a04f81">Majorant:</a></li> -<li><a href="#org84bdb2d">Minorant:</a></li> -<li><a href="#orgcba58a3">Borne supérieure:</a></li> -<li><a href="#org38b79f1">Borne inférieure:</a></li> -<li><a href="#orgb36c17b">Maximum :</a></li> -<li><a href="#orgfe7719f">Minimum :</a></li> -<li><a href="#org15af67d">Remarques :</a></li> +<li><a href="#orgb6fd133">Majorant:</a></li> +<li><a href="#org186a70c">Minorant:</a></li> +<li><a href="#org64b766d">Borne supérieure:</a></li> +<li><a href="#org11d4c50">Borne inférieure:</a></li> +<li><a href="#orge205f5a">Maximum :</a></li> +<li><a href="#org1afad1e">Minimum :</a></li> +<li><a href="#orge3a6538">Remarques :</a></li> </ul> </li> </ul> </li> -<li><a href="#org49b0544">3rd cours :Les suites numériques <i>Oct 5</i> :</a> +<li><a href="#org286c633">3rd cours :Les suites numériques <i>Oct 5</i> :</a> <ul> <li> <ul> -<li><a href="#org73b66a2">Définition :</a></li> -<li><a href="#org865ba25">Définition N°2 :</a></li> +<li><a href="#org2a95022">Définition :</a></li> +<li><a href="#org1bec8d7">Définition N°2 :</a></li> </ul> </li> -<li><a href="#org67fae0d">Opérations sur les suites :</a> +<li><a href="#orgf0c88cd">Opérations sur les suites :</a> <ul> -<li><a href="#org6de1660">La somme :</a></li> -<li><a href="#orgda301a2">Le produit :</a></li> -<li><a href="#org92603b8">Inverse d’une suite :</a></li> -<li><a href="#org0e9a26c">Produit d’une suite par un scalaire :</a></li> +<li><a href="#org2163510">La somme :</a></li> +<li><a href="#org1ec5af6">Le produit :</a></li> +<li><a href="#orgbdb350d">Inverse d’une suite :</a></li> +<li><a href="#orge5cf6d9">Produit d’une suite par un scalaire :</a></li> </ul> </li> -<li><a href="#org5f93512">Suite bornée :</a></li> -<li><a href="#org940ba0d">Suite majorée :</a></li> -<li><a href="#org9f967d2">Suite minorée :</a></li> -<li><a href="#org8ff5480">Suites monotones :</a> +<li><a href="#org2bab5af">Suite bornée :</a></li> +<li><a href="#org8aa293b">Suite majorée :</a></li> +<li><a href="#org83e8a37">Suite minorée :</a></li> +<li><a href="#org4a078cc">Suites monotones :</a> <ul> -<li><a href="#orgce608f3">Les suites croissantes :</a></li> -<li><a href="#orgb6caa62">Les suites décroissantes :</a></li> +<li><a href="#orgeb783d1">Les suites croissantes :</a></li> +<li><a href="#orgcc61cbf">Les suites décroissantes :</a></li> </ul> </li> </ul> </li> -<li><a href="#org5b5bade">Série TD N°1 : <i>Oct 6</i></a> +<li><a href="#org9be42e0">Série TD N°1 : <i>Oct 6</i></a> <ul> -<li><a href="#org16a2d89">Exo 1 :</a> +<li><a href="#orgd3e58b6">Exo 1 :</a> <ul> -<li><a href="#org145fce2">Ensemble A :</a></li> -<li><a href="#org539689f">Ensemble B :</a></li> -<li><a href="#orgced67f0">Ensemble C :</a></li> -<li><a href="#orgedfa37a">Ensemble D :</a></li> -<li><a href="#org13b9ce0">Ensemble E :</a></li> +<li><a href="#org0087dc5">Ensemble A :</a></li> +<li><a href="#org4c859a3">Ensemble B :</a></li> +<li><a href="#org2ad9bb3">Ensemble C :</a></li> +<li><a href="#orgde49b00">Ensemble D :</a></li> +<li><a href="#orgb857fc5">Ensemble E :</a></li> </ul> </li> -<li><a href="#org227cd11">Exo 2 :</a> +<li><a href="#org3241c28">Exo 2 :</a> <ul> -<li><a href="#org1c6a24b">Ensemble A :</a></li> -<li><a href="#org51c6bfc">Ensemble B :</a></li> -<li><a href="#org7accf6a">Ensemble C :</a></li> -<li><a href="#org4f20a91">Ensemble D :</a></li> -<li><a href="#org9a20270">Ensemble E :</a></li> +<li><a href="#org77ffa3e">Ensemble A :</a></li> +<li><a href="#org151d601">Ensemble B :</a></li> +<li><a href="#orgbc1efd9">Ensemble C :</a></li> +<li><a href="#org0eda8d2">Ensemble D :</a></li> +<li><a href="#org9b9b691">Ensemble E :</a></li> </ul> </li> -<li><a href="#orgae3b875">Exo 3 :</a> +<li><a href="#org36dc1da">Exo 3 :</a> <ul> -<li><a href="#org261f974">Question 1 :</a></li> -<li><a href="#orge6e1ce3">Question 2 :</a></li> +<li><a href="#org7999092">Question 1 :</a></li> +<li><a href="#orgb9f7a15">Question 2 :</a></li> </ul> </li> </ul> </li> -<li><a href="#orge45c9b7">4th cours (Suite) : <i>Oct 10</i></a> +<li><a href="#org3da135e">4th cours (Suite) : <i>Oct 10</i></a> <ul> -<li><a href="#org58b09d5">Les suites convergentes</a> +<li><a href="#org639877a">Les suites convergentes</a> <ul> -<li><a href="#org820f474">Remarque :</a></li> +<li><a href="#orgce5e8f7">Remarque :</a></li> </ul> </li> -<li><a href="#orge043bff">Theoreme d’encadrement</a></li> -<li><a href="#org8460bc2">Suites arithmetiques</a> +<li><a href="#orga659f1f">Theoreme d’encadrement</a></li> +<li><a href="#org4c1ed41">Suites arithmetiques</a> <ul> -<li><a href="#org51b313a">Forme general</a></li> -<li><a href="#org63ac12f">Somme des n premiers termes</a></li> +<li><a href="#org5b887fd">Forme general</a></li> +<li><a href="#orgbd36410">Somme des n premiers termes</a></li> </ul> </li> -<li><a href="#org24bc205">Suites géométriques</a> +<li><a href="#orge060a6b">Suites géométriques</a> <ul> -<li><a href="#orgc70d4f0">Forme general</a></li> -<li><a href="#org276a52e">Somme des n premiers termes</a></li> +<li><a href="#org7eb64b7">Forme general</a></li> +<li><a href="#org4a1c78c">Somme des n premiers termes</a></li> </ul> </li> </ul> </li> -<li><a href="#orgb4ceb77">5th cours (suite) : <i>Oct 12</i></a> +<li><a href="#org9ad98cf">5th cours (suite) : <i>Oct 12</i></a> <ul> -<li><a href="#org5f6fa60">Suites adjacentes:</a></li> -<li><a href="#orgea8a031">Suites extraites (sous-suites):</a> +<li><a href="#org3ef59f2">Suites adjacentes:</a></li> +<li><a href="#org05716a0">Suites extraites (sous-suites):</a> <ul> -<li><a href="#org0a4213f">Remarques:</a></li> +<li><a href="#org312cfda">Remarques:</a></li> </ul> </li> -<li><a href="#orgec23ceb">Suites de Cauchy:</a> +<li><a href="#orgbfa31ac">Suites de Cauchy:</a> <ul> -<li><a href="#org04bdc9a">Remarque :</a></li> +<li><a href="#org60c9452">Remarque :</a></li> </ul> </li> -<li><a href="#orgc639b18">Théorème de Bolzano Weirstrass:</a></li> +<li><a href="#org678d2ef">Théorème de Bolzano Weirstrass:</a></li> </ul> </li> </ul> </div> </div> -<div id="outline-container-org69b61eb" class="outline-2"> -<h2 id="org69b61eb">Contenu de la Matiére</h2> -<div class="outline-text-2" id="text-org69b61eb"> +<div id="outline-container-org96a1915" class="outline-2"> +<h2 id="org96a1915">Contenu de la Matiére</h2> +<div class="outline-text-2" id="text-org96a1915"> </div> -<div id="outline-container-org0a8600d" class="outline-3"> -<h3 id="org0a8600d">Chapitre 1 : Quelque propriétés de ℝ</h3> -<div class="outline-text-3" id="text-org0a8600d"> +<div id="outline-container-org92a5c1e" class="outline-3"> +<h3 id="org92a5c1e">Chapitre 1 : Quelque propriétés de ℝ</h3> +<div class="outline-text-3" id="text-org92a5c1e"> <ul class="org-ul"> <li>Structure algébrique de ℝ<br /></li> <li>L’ordre dans ℝ<br /></li> @@ -201,9 +201,9 @@ </ul> </div> </div> -<div id="outline-container-org85fd1b6" class="outline-3"> -<h3 id="org85fd1b6">Chapitre 2 : Les suites numériques réelles</h3> -<div class="outline-text-3" id="text-org85fd1b6"> +<div id="outline-container-org43abd6f" class="outline-3"> +<h3 id="org43abd6f">Chapitre 2 : Les suites numériques réelles</h3> +<div class="outline-text-3" id="text-org43abd6f"> <ul class="org-ul"> <li>Définition : convergence, opérations sur les suites convergentes<br /></li> <li>Theoréme de convergence, Theoréme de <span class="underline">_</span> suites, sans suites, extension au limites infinies<br /></li> @@ -211,9 +211,9 @@ </ul> </div> </div> -<div id="outline-container-org0f00d84" class="outline-3"> -<h3 id="org0f00d84">Chapitre 3 : Limites et continuité des fonctions réelles d’une variable réelle</h3> -<div class="outline-text-3" id="text-org0f00d84"> +<div id="outline-container-orgb7dbd4d" class="outline-3"> +<h3 id="orgb7dbd4d">Chapitre 3 : Limites et continuité des fonctions réelles d’une variable réelle</h3> +<div class="outline-text-3" id="text-orgb7dbd4d"> <ul class="org-ul"> <li>Les limites : définition, opérations sur les limites, les formes inditerminées<br /></li> <li>La continuité : définition, Theorémes fondamentaux<br /></li> @@ -221,17 +221,17 @@ </ul> </div> </div> -<div id="outline-container-org65466d0" class="outline-3"> -<h3 id="org65466d0">Chapitre 4 : La dérivabilité et son interprétation géometrique</h3> -<div class="outline-text-3" id="text-org65466d0"> +<div id="outline-container-orgb39022d" class="outline-3"> +<h3 id="orgb39022d">Chapitre 4 : La dérivabilité et son interprétation géometrique</h3> +<div class="outline-text-3" id="text-orgb39022d"> <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<br /></li> </ul> </div> </div> -<div id="outline-container-org12481b8" class="outline-3"> -<h3 id="org12481b8">Chapitre 5 : Les fonctions trigonométriques réciproques, fonctions hypérboliques réciproques</h3> -<div class="outline-text-3" id="text-org12481b8"> +<div id="outline-container-orgbfa8dc6" class="outline-3"> +<h3 id="orgbfa8dc6">Chapitre 5 : Les fonctions trigonométriques réciproques, fonctions hypérboliques réciproques</h3> +<div class="outline-text-3" id="text-orgbfa8dc6"> <ul class="org-ul"> <li>Comparaison asymptotique<br /></li> <li>Symbole de lamdau (lambda ?), et notions des fonctions équivalentes<br /></li> @@ -242,13 +242,13 @@ </div> </div> </div> -<div id="outline-container-org3fc2976" class="outline-2"> -<h2 id="org3fc2976">Premier cours : Quelque propriétés de ℝ <i>Sep 26</i> :</h2> -<div class="outline-text-2" id="text-org3fc2976"> +<div id="outline-container-org0e8210e" class="outline-2"> +<h2 id="org0e8210e">Premier cours : Quelque propriétés de ℝ <i>Sep 26</i> :</h2> +<div class="outline-text-2" id="text-org0e8210e"> </div> -<div id="outline-container-org4e71cb9" class="outline-3"> -<h3 id="org4e71cb9">La loi de composition interne dans E :</h3> -<div class="outline-text-3" id="text-org4e71cb9"> +<div id="outline-container-org4b5ef0e" class="outline-3"> +<h3 id="org4b5ef0e">La loi de composition interne dans E :</h3> +<div class="outline-text-3" id="text-org4b5ef0e"> <p> @ : E x E —> E<br /> (x,y) —> x @ y<br /> @@ -262,9 +262,9 @@ <b>∀ x,y ε E</b><br /> </p> </div> -<div id="outline-container-orgbc38c17" class="outline-4"> -<h4 id="orgbc38c17"><b>Example : Addition</b></h4> -<div class="outline-text-4" id="text-orgbc38c17"> +<div id="outline-container-org26ad93c" class="outline-4"> +<h4 id="org26ad93c"><b>Example : Addition</b></h4> +<div class="outline-text-4" id="text-org26ad93c"> <p> Est ce que l’addition (+) est L.C.I dans ℕ ?<br /> </p> @@ -286,9 +286,9 @@ Donc : + est L.C.I dans ℕ<br /> </p> </div> </div> -<div id="outline-container-org32a22fa" class="outline-4"> -<h4 id="org32a22fa"><b>Example : soustraction</b></h4> -<div class="outline-text-4" id="text-org32a22fa"> +<div id="outline-container-org716f99d" class="outline-4"> +<h4 id="org716f99d"><b>Example : soustraction</b></h4> +<div class="outline-text-4" id="text-org716f99d"> <p> Est ce que la soustraction (-) est L.C.I dans ℕ?<br /> </p> @@ -308,9 +308,9 @@ Est ce que la soustraction (-) est L.C.I dans ℕ?<br /> </div> </div> </div> -<div id="outline-container-org19f1c04" class="outline-3"> -<h3 id="org19f1c04">La loi de composition externe dans E :</h3> -<div class="outline-text-3" id="text-org19f1c04"> +<div id="outline-container-orgcd239ec" class="outline-3"> +<h3 id="orgcd239ec">La loi de composition externe dans E :</h3> +<div class="outline-text-3" id="text-orgcd239ec"> <p> @ est L.C.E dans E, K est un corps<br /> </p> @@ -328,9 +328,9 @@ K x E —> E<br /> </p> </div> </div> -<div id="outline-container-org24ed87c" class="outline-3"> -<h3 id="org24ed87c">Groupes :</h3> -<div class="outline-text-3" id="text-org24ed87c"> +<div id="outline-container-org6aa8256" class="outline-3"> +<h3 id="org6aa8256">Groupes :</h3> +<div class="outline-text-3" id="text-org6aa8256"> <p> <i>Soit E un ensemble, soit @ une L.C.I dans E</i><br /> </p> @@ -339,9 +339,9 @@ K x E —> E<br /> (E, @) est un groupe Si :<br /> </p> </div> -<div id="outline-container-org8f69286" class="outline-4"> -<h4 id="org8f69286">Il contiens un élement neutre</h4> -<div class="outline-text-4" id="text-org8f69286"> +<div id="outline-container-org3a88117" class="outline-4"> +<h4 id="org3a88117">Il contiens un élement neutre</h4> +<div class="outline-text-4" id="text-org3a88117"> <p> ∀ x ∈ E ; ∃ e ∈ E<br /> </p> @@ -359,9 +359,9 @@ On appelle <b>e</b> élement neutre<br /> </p> </div> </div> -<div id="outline-container-org6747a03" class="outline-4"> -<h4 id="org6747a03">Il contiens un élément symétrique</h4> -<div class="outline-text-4" id="text-org6747a03"> +<div id="outline-container-orgf17cd87" class="outline-4"> +<h4 id="orgf17cd87">Il contiens un élément symétrique</h4> +<div class="outline-text-4" id="text-orgf17cd87"> <p> ∀ x ∈ E ; ∃ x’ ∈ E ; x @ x’ = x’ @ x = e<br /> </p> @@ -383,9 +383,9 @@ On appelle <b>x’</b> élèment symétrique<br /> </p> </div> </div> -<div id="outline-container-org4bf37c2" class="outline-4"> -<h4 id="org4bf37c2">@ est cummutative :</h4> -<div class="outline-text-4" id="text-org4bf37c2"> +<div id="outline-container-org541a8aa" class="outline-4"> +<h4 id="org541a8aa">@ est cummutative :</h4> +<div class="outline-text-4" id="text-org541a8aa"> <p> ∀ (x , x’) ∈ E x E ; x @ x’ = x’ @ x<br /> </p> @@ -396,19 +396,19 @@ On appelle <b>x’</b> élèment symétrique<br /> </div> </div> </div> -<div id="outline-container-orgd89b7dc" class="outline-3"> -<h3 id="orgd89b7dc">Anneaux :</h3> -<div class="outline-text-3" id="text-orgd89b7dc"> +<div id="outline-container-org325ac76" class="outline-3"> +<h3 id="org325ac76">Anneaux :</h3> +<div class="outline-text-3" id="text-org325ac76"> <p> Soit E un ensemble, (E , @ , !) est un anneau si :<br /> </p> </div> -<div id="outline-container-org0974b29" class="outline-4"> -<h4 id="org0974b29">(E ; @) est un groupe cummutatif</h4> +<div id="outline-container-orgb12d61b" class="outline-4"> +<h4 id="orgb12d61b">(E ; @) est un groupe cummutatif</h4> </div> -<div id="outline-container-org9b00b5e" class="outline-4"> -<h4 id="org9b00b5e">! est une loi associative :</h4> -<div class="outline-text-4" id="text-org9b00b5e"> +<div id="outline-container-org2f6c910" class="outline-4"> +<h4 id="org2f6c910">! est une loi associative :</h4> +<div class="outline-text-4" id="text-org2f6c910"> <p> ∀ x , y , z ∈ E<br /> </p> @@ -418,9 +418,9 @@ Soit E un ensemble, (E , @ , !) est un anneau si :<br /> </p> </div> </div> -<div id="outline-container-org248e491" class="outline-4"> -<h4 id="org248e491">Distribution de ! par rapport à @ :</h4> -<div class="outline-text-4" id="text-org248e491"> +<div id="outline-container-org288714a" class="outline-4"> +<h4 id="org288714a">Distribution de ! par rapport à @ :</h4> +<div class="outline-text-4" id="text-org288714a"> <p> ∀ x , y , z ∈ E<br /> </p> @@ -430,33 +430,33 @@ Soit E un ensemble, (E , @ , !) est un anneau si :<br /> </p> </div> </div> -<div id="outline-container-org6cbe38c" class="outline-4"> -<h4 id="org6cbe38c">L’existance d’un élèment neutre de ! :</h4> -<div class="outline-text-4" id="text-org6cbe38c"> +<div id="outline-container-org92b8438" class="outline-4"> +<h4 id="org92b8438">L’existance d’un élèment neutre de ! :</h4> +<div class="outline-text-4" id="text-org92b8438"> <p> ∀ x ∈ E , ∃ e ∈ E , x ! e = e ! x = x<br /> </p> </div> </div> -<div id="outline-container-orgd1b5bee" class="outline-4"> -<h4 id="orgd1b5bee">! est cummutative :</h4> -<div class="outline-text-4" id="text-orgd1b5bee"> +<div id="outline-container-org12cac33" class="outline-4"> +<h4 id="org12cac33">! est cummutative :</h4> +<div class="outline-text-4" id="text-org12cac33"> <p> ∀ x , y ∈ E , x ! y = y ! x<br /> </p> </div> </div> </div> -<div id="outline-container-org7fdacd4" class="outline-3"> -<h3 id="org7fdacd4">Corps :</h3> -<div class="outline-text-3" id="text-org7fdacd4"> +<div id="outline-container-orgbdac47c" class="outline-3"> +<h3 id="orgbdac47c">Corps :</h3> +<div class="outline-text-3" id="text-orgbdac47c"> <p> (E , @ , !) est un corps si les 5 conditions en haut sont vérifiées + cette condition :<br /> </p> </div> -<div id="outline-container-org3853ce7" class="outline-4"> -<h4 id="org3853ce7">La symétrie :</h4> -<div class="outline-text-4" id="text-org3853ce7"> +<div id="outline-container-orgea3147a" class="outline-4"> +<h4 id="orgea3147a">La symétrie :</h4> +<div class="outline-text-4" id="text-orgea3147a"> <p> ∀ x ∈ E ; ∃ x’ ∈ E , x ! x’ = x’ ! x = e<br /> </p> @@ -468,13 +468,13 @@ x’ est l’élément symétrique de x par rapport à !<br /> </div> </div> </div> -<div id="outline-container-org3922e65" class="outline-3"> -<h3 id="org3922e65">Exercice : (ℝ, +, x) corps ou pas ?</h3> -<div class="outline-text-3" id="text-org3922e65"> +<div id="outline-container-org012a6fe" class="outline-3"> +<h3 id="org012a6fe">Exercice : (ℝ, +, x) corps ou pas ?</h3> +<div class="outline-text-3" id="text-org012a6fe"> </div> -<div id="outline-container-org13f7b7e" class="outline-4"> -<h4 id="org13f7b7e">Est-ce un Anneau ?</h4> -<div class="outline-text-4" id="text-org13f7b7e"> +<div id="outline-container-org0934303" class="outline-4"> +<h4 id="org0934303">Est-ce un Anneau ?</h4> +<div class="outline-text-4" id="text-org0934303"> <ul class="org-ul"> <li>(ℝ, +) est un groupe commutatif<br /></li> <li>x est une loi associative : (a x b) x c = a x (b x c)<br /></li> @@ -488,9 +488,9 @@ Oui c’est un anneau<br /> </p> </div> </div> -<div id="outline-container-org90e2795" class="outline-4"> -<h4 id="org90e2795">Est-ce un corps ?</h4> -<div class="outline-text-4" id="text-org90e2795"> +<div id="outline-container-orgb7020c8" class="outline-4"> +<h4 id="orgb7020c8">Est-ce un corps ?</h4> +<div class="outline-text-4" id="text-orgb7020c8"> <ul class="org-ul"> <li>Oui : ∀ x ∈ ℝ\{e} ; x * x’ = 1<br /></li> </ul> @@ -498,13 +498,13 @@ Oui c’est un anneau<br /> </div> </div> </div> -<div id="outline-container-org76e7ef3" class="outline-2"> -<h2 id="org76e7ef3">2nd cours :L’ordre dans ℝ, Majorant, minorant, borne superieure, borne inférieure <i>Oct 3</i> :</h2> -<div class="outline-text-2" id="text-org76e7ef3"> +<div id="outline-container-orgabdfea2" class="outline-2"> +<h2 id="orgabdfea2">2nd cours :L’ordre dans ℝ, Majorant, minorant, borne superieure, borne inférieure <i>Oct 3</i> :</h2> +<div class="outline-text-2" id="text-orgabdfea2"> </div> -<div id="outline-container-org94449fa" class="outline-3"> -<h3 id="org94449fa">L’ordre dans ℝ</h3> -<div class="outline-text-3" id="text-org94449fa"> +<div id="outline-container-org30b59ae" class="outline-3"> +<h3 id="org30b59ae">L’ordre dans ℝ</h3> +<div class="outline-text-3" id="text-org30b59ae"> <p> (ℝ, +, x) est un corps, Soit R une relation d’ordre dans ℝ si :<br /> </p> @@ -530,13 +530,13 @@ R est reflexive :<br /> ∀ x, y, z ∈ ℝ , (x R y and y R z) ⇒ x R z<br /></li> </ol> </div> -<div id="outline-container-org17e010d" class="outline-4"> -<h4 id="org17e010d">Exemples :</h4> -<div class="outline-text-4" id="text-org17e010d"> +<div id="outline-container-org86a0035" class="outline-4"> +<h4 id="org86a0035">Exemples :</h4> +<div class="outline-text-4" id="text-org86a0035"> </div> <ul class="org-ul"> -<li><a id="org7ca8c24"></a>Exemple numéro 1:<br /> -<div class="outline-text-5" id="text-org7ca8c24"> +<li><a id="orgeaa24ca"></a>Exemple numéro 1:<br /> +<div class="outline-text-5" id="text-orgeaa24ca"> <p> (ℝ , +, x) est un corps. Est ce la relation < est une relation d’ordre dans ℝ ?<br /> </p> @@ -547,8 +547,8 @@ Non, pourquoi ? parce que elle est pas réflexive : ∀ x ∈ ℝ, x < x <b>< </p> </div> </li> -<li><a id="org2160b00"></a>Exemple numéro 2:<br /> -<div class="outline-text-5" id="text-org2160b00"> +<li><a id="org13e92d7"></a>Exemple numéro 2:<br /> +<div class="outline-text-5" id="text-org13e92d7"> <p> (ℝ , +, x) est un corps. Est ce la relation ≥ est une relation d’ordre dans ℝ ?<br /> </p> @@ -563,13 +563,13 @@ Non, pourquoi ? parce que elle est pas réflexive : ∀ x ∈ ℝ, x < x <b>< </ul> </div> </div> -<div id="outline-container-orga26a211" class="outline-3"> -<h3 id="orga26a211">Majorant, minorant, borne supérieure, borne inférieure</h3> -<div class="outline-text-3" id="text-orga26a211"> +<div id="outline-container-org43ad665" class="outline-3"> +<h3 id="org43ad665">Majorant, minorant, borne supérieure, borne inférieure</h3> +<div class="outline-text-3" id="text-org43ad665"> </div> -<div id="outline-container-org3a04f81" class="outline-4"> -<h4 id="org3a04f81">Majorant:</h4> -<div class="outline-text-4" id="text-org3a04f81"> +<div id="outline-container-orgb6fd133" class="outline-4"> +<h4 id="orgb6fd133">Majorant:</h4> +<div class="outline-text-4" id="text-orgb6fd133"> <p> Soit E un sous-ensemble de ℝ (E ⊆ ℝ)<br /> </p> @@ -580,9 +580,9 @@ Soit a ∈ ℝ, a est un majorant de E Si :∀ x ∈ E , x ≤ a<br /> </p> </div> </div> -<div id="outline-container-org84bdb2d" class="outline-4"> -<h4 id="org84bdb2d">Minorant:</h4> -<div class="outline-text-4" id="text-org84bdb2d"> +<div id="outline-container-org186a70c" class="outline-4"> +<h4 id="org186a70c">Minorant:</h4> +<div class="outline-text-4" id="text-org186a70c"> <p> Soit E un sous-ensemble de ℝ (E ⊆ ℝ)<br /> </p> @@ -593,41 +593,41 @@ Soit b ∈ ℝ, b est un minorant de E Si :∀ x ∈ E , x ≥ b<br /> </p> </div> </div> -<div id="outline-container-orgcba58a3" class="outline-4"> -<h4 id="orgcba58a3">Borne supérieure:</h4> -<div class="outline-text-4" id="text-orgcba58a3"> +<div id="outline-container-org64b766d" class="outline-4"> +<h4 id="org64b766d">Borne supérieure:</h4> +<div class="outline-text-4" id="text-org64b766d"> <p> La borne supérieure est le plus petit des majorants <i>Sup(E) = Borne supérieure</i><br /> </p> </div> </div> -<div id="outline-container-org38b79f1" class="outline-4"> -<h4 id="org38b79f1">Borne inférieure:</h4> -<div class="outline-text-4" id="text-org38b79f1"> +<div id="outline-container-org11d4c50" class="outline-4"> +<h4 id="org11d4c50">Borne inférieure:</h4> +<div class="outline-text-4" id="text-org11d4c50"> <p> La borne inférieure est le plus grand des minorant <i>Inf(E) = Borne inférieure</i><br /> </p> </div> </div> -<div id="outline-container-orgb36c17b" class="outline-4"> -<h4 id="orgb36c17b">Maximum :</h4> -<div class="outline-text-4" id="text-orgb36c17b"> +<div id="outline-container-orge205f5a" class="outline-4"> +<h4 id="orge205f5a">Maximum :</h4> +<div class="outline-text-4" id="text-orge205f5a"> <p> E ⊆ ℝ, a est un maximum de E (Max(E)) Si : a ∈ E ; ∀x ∈ E, x ≤ a.<br /> </p> </div> </div> -<div id="outline-container-orgfe7719f" class="outline-4"> -<h4 id="orgfe7719f">Minimum :</h4> -<div class="outline-text-4" id="text-orgfe7719f"> +<div id="outline-container-org1afad1e" class="outline-4"> +<h4 id="org1afad1e">Minimum :</h4> +<div class="outline-text-4" id="text-org1afad1e"> <p> E ⊆ ℝ, b est un minimum de E (Min(E)) Si : b ∈ E ; ∀x ∈ E, x ≥ b.<br /> </p> </div> </div> -<div id="outline-container-org15af67d" class="outline-4"> -<h4 id="org15af67d">Remarques :</h4> -<div class="outline-text-4" id="text-org15af67d"> +<div id="outline-container-orge3a6538" class="outline-4"> +<h4 id="orge3a6538">Remarques :</h4> +<div class="outline-text-4" id="text-orge3a6538"> <p> A et B deux ensembles bornés (Minoré et Majoré) :<br /> </p> @@ -643,13 +643,13 @@ A et B deux ensembles bornés (Minoré et Majoré) :<br /> </div> </div> </div> -<div id="outline-container-org49b0544" class="outline-2"> -<h2 id="org49b0544">3rd cours :Les suites numériques <i>Oct 5</i> :</h2> -<div class="outline-text-2" id="text-org49b0544"> +<div id="outline-container-org286c633" class="outline-2"> +<h2 id="org286c633">3rd cours :Les suites numériques <i>Oct 5</i> :</h2> +<div class="outline-text-2" id="text-org286c633"> </div> -<div id="outline-container-org73b66a2" class="outline-4"> -<h4 id="org73b66a2">Définition :</h4> -<div class="outline-text-4" id="text-org73b66a2"> +<div id="outline-container-org2a95022" class="outline-4"> +<h4 id="org2a95022">Définition :</h4> +<div class="outline-text-4" id="text-org2a95022"> <p> Soit (Un)n ∈ ℕ une suite numérique , (Un)n est une application de ℕ dans ℝ:<br /> </p> @@ -670,8 +670,8 @@ n -—> U(n) = Un<br /> </ol> </div> <ul class="org-ul"> -<li><a id="orgba0d8c3"></a>Exemple :<br /> -<div class="outline-text-6" id="text-orgba0d8c3"> +<li><a id="org4364064"></a>Exemple :<br /> +<div class="outline-text-6" id="text-org4364064"> <p> U : ℕ* -—> ℝ<br /> </p> @@ -689,16 +689,16 @@ n -—> 1/n<br /> </li> </ul> </div> -<div id="outline-container-org865ba25" class="outline-4"> -<h4 id="org865ba25">Définition N°2 :</h4> -<div class="outline-text-4" id="text-org865ba25"> +<div id="outline-container-org1bec8d7" class="outline-4"> +<h4 id="org1bec8d7">Définition N°2 :</h4> +<div class="outline-text-4" id="text-org1bec8d7"> <p> On peut définir une suite â partir d’une relation de récurrence entre deux termes successifs et le premier terme.<br /> </p> </div> <ul class="org-ul"> -<li><a id="org0758a65"></a>Exemple :<br /> -<div class="outline-text-6" id="text-org0758a65"> +<li><a id="org0fd87c4"></a>Exemple :<br /> +<div class="outline-text-6" id="text-org0fd87c4"> <p> U(n+1) = Un /2<br /> </p> @@ -711,37 +711,37 @@ U(1)= 1<br /> </li> </ul> </div> -<div id="outline-container-org67fae0d" class="outline-3"> -<h3 id="org67fae0d">Opérations sur les suites :</h3> -<div class="outline-text-3" id="text-org67fae0d"> +<div id="outline-container-orgf0c88cd" class="outline-3"> +<h3 id="orgf0c88cd">Opérations sur les suites :</h3> +<div class="outline-text-3" id="text-orgf0c88cd"> </div> -<div id="outline-container-org6de1660" class="outline-4"> -<h4 id="org6de1660">La somme :</h4> -<div class="outline-text-4" id="text-org6de1660"> +<div id="outline-container-org2163510" class="outline-4"> +<h4 id="org2163510">La somme :</h4> +<div class="outline-text-4" id="text-org2163510"> <p> Soient (Un) et (Vn) deux suites, la somme de (Un) et (Vn) est une suite de terme général Un + Vn<br /> </p> </div> </div> -<div id="outline-container-orgda301a2" class="outline-4"> -<h4 id="orgda301a2">Le produit :</h4> -<div class="outline-text-4" id="text-orgda301a2"> +<div id="outline-container-org1ec5af6" class="outline-4"> +<h4 id="org1ec5af6">Le produit :</h4> +<div class="outline-text-4" id="text-org1ec5af6"> <p> Soient (Un)n et (Vn)n deux suites alors (Un) x (Vn) est une autre suite de terme général Un x Vn<br /> </p> </div> </div> -<div id="outline-container-org92603b8" class="outline-4"> -<h4 id="org92603b8">Inverse d’une suite :</h4> -<div class="outline-text-4" id="text-org92603b8"> +<div id="outline-container-orgbdb350d" class="outline-4"> +<h4 id="orgbdb350d">Inverse d’une suite :</h4> +<div class="outline-text-4" id="text-orgbdb350d"> <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<br /> </p> </div> </div> -<div id="outline-container-org0e9a26c" class="outline-4"> -<h4 id="org0e9a26c">Produit d’une suite par un scalaire :</h4> -<div class="outline-text-4" id="text-org0e9a26c"> +<div id="outline-container-orge5cf6d9" class="outline-4"> +<h4 id="orge5cf6d9">Produit d’une suite par un scalaire :</h4> +<div class="outline-text-4" id="text-orge5cf6d9"> <p> Soit (Un) une suite de T.G Un<br /> </p> @@ -753,17 +753,17 @@ Soit (Un) une suite de T.G Un<br /> </div> </div> </div> -<div id="outline-container-org5f93512" class="outline-3"> -<h3 id="org5f93512">Suite bornée :</h3> -<div class="outline-text-3" id="text-org5f93512"> +<div id="outline-container-org2bab5af" class="outline-3"> +<h3 id="org2bab5af">Suite bornée :</h3> +<div class="outline-text-3" id="text-org2bab5af"> <p> Une suite (Un) est bornée si (Un) majorée et minorée<br /> </p> </div> </div> -<div id="outline-container-org940ba0d" class="outline-3"> -<h3 id="org940ba0d">Suite majorée :</h3> -<div class="outline-text-3" id="text-org940ba0d"> +<div id="outline-container-org8aa293b" class="outline-3"> +<h3 id="org8aa293b">Suite majorée :</h3> +<div class="outline-text-3" id="text-org8aa293b"> <p> Soit (Un) une suite<br /> </p> @@ -774,9 +774,9 @@ U : (Un) est majorée par M ∈ ℝ ; ∀ n ∈ ℕ ; ∃ M ∈ ℝ , Un ≤ M<b </p> </div> </div> -<div id="outline-container-org9f967d2" class="outline-3"> -<h3 id="org9f967d2">Suite minorée :</h3> -<div class="outline-text-3" id="text-org9f967d2"> +<div id="outline-container-org83e8a37" class="outline-3"> +<h3 id="org83e8a37">Suite minorée :</h3> +<div class="outline-text-3" id="text-org83e8a37"> <p> Soit (Un) une suite<br /> </p> @@ -787,13 +787,13 @@ U : (Un) est minorée par M ∈ ℝ ; ∀ n ∈ ℕ ; ∃ M ∈ ℝ , Un ≥ M<b </p> </div> </div> -<div id="outline-container-org8ff5480" class="outline-3"> -<h3 id="org8ff5480">Suites monotones :</h3> -<div class="outline-text-3" id="text-org8ff5480"> +<div id="outline-container-org4a078cc" class="outline-3"> +<h3 id="org4a078cc">Suites monotones :</h3> +<div class="outline-text-3" id="text-org4a078cc"> </div> -<div id="outline-container-orgce608f3" class="outline-4"> -<h4 id="orgce608f3">Les suites croissantes :</h4> -<div class="outline-text-4" id="text-orgce608f3"> +<div id="outline-container-orgeb783d1" class="outline-4"> +<h4 id="orgeb783d1">Les suites croissantes :</h4> +<div class="outline-text-4" id="text-orgeb783d1"> <p> Soit (Un)n est une suite<br /> </p> @@ -804,9 +804,9 @@ Soit (Un)n est une suite<br /> </p> </div> </div> -<div id="outline-container-orgb6caa62" class="outline-4"> -<h4 id="orgb6caa62">Les suites décroissantes :</h4> -<div class="outline-text-4" id="text-orgb6caa62"> +<div id="outline-container-orgcc61cbf" class="outline-4"> +<h4 id="orgcc61cbf">Les suites décroissantes :</h4> +<div class="outline-text-4" id="text-orgcc61cbf"> <p> Soit (Un)n est une suite<br /> </p> @@ -819,45 +819,45 @@ Soit (Un)n est une suite<br /> </div> </div> </div> -<div id="outline-container-org5b5bade" class="outline-2"> -<h2 id="org5b5bade">Série TD N°1 : <i>Oct 6</i></h2> -<div class="outline-text-2" id="text-org5b5bade"> +<div id="outline-container-org9be42e0" class="outline-2"> +<h2 id="org9be42e0">Série TD N°1 : <i>Oct 6</i></h2> +<div class="outline-text-2" id="text-org9be42e0"> </div> -<div id="outline-container-org16a2d89" class="outline-3"> -<h3 id="org16a2d89">Exo 1 :</h3> -<div class="outline-text-3" id="text-org16a2d89"> +<div id="outline-container-orgd3e58b6" class="outline-3"> +<h3 id="orgd3e58b6">Exo 1 :</h3> +<div class="outline-text-3" id="text-orgd3e58b6"> </div> -<div id="outline-container-org145fce2" class="outline-4"> -<h4 id="org145fce2">Ensemble A :</h4> -<div class="outline-text-4" id="text-org145fce2"> +<div id="outline-container-org0087dc5" class="outline-4"> +<h4 id="org0087dc5">Ensemble A :</h4> +<div class="outline-text-4" id="text-org0087dc5"> <p> A = {-1/n , n ∈ ℕ *}<br /> </p> </div> <ul class="org-ul"> -<li><a id="orgf6179c5"></a>Borne inférieure<br /> -<div class="outline-text-5" id="text-orgf6179c5"> +<li><a id="org0b6fb26"></a>Borne inférieure<br /> +<div class="outline-text-5" id="text-org0b6fb26"> <p> ∀ n ∈ ℕ* , -1/n ≥ -1 . -1 est la borne inférieure de l’ensemble A<br /> </p> </div> </li> -<li><a id="orgae45bd6"></a>Minimum :<br /> -<div class="outline-text-5" id="text-orgae45bd6"> +<li><a id="org62dc78e"></a>Minimum :<br /> +<div class="outline-text-5" id="text-org62dc78e"> <p> ∀ n ∈ ℕ* , -1/n ≥ -1 . -1 est le Minimum de l’ensemble A<br /> </p> </div> </li> -<li><a id="orgc7c3433"></a>Borne supérieure :<br /> -<div class="outline-text-5" id="text-orgc7c3433"> +<li><a id="orgf29cc66"></a>Borne supérieure :<br /> +<div class="outline-text-5" id="text-orgf29cc66"> <p> ∀ n ∈ ℕ* , -1/n ≤ 0 . 0 est la borne supérieure de l’ensemble A<br /> </p> </div> </li> -<li><a id="orgab09e1c"></a>Maximum :<br /> -<div class="outline-text-5" id="text-orgab09e1c"> +<li><a id="org754a088"></a>Maximum :<br /> +<div class="outline-text-5" id="text-org754a088"> <p> L’ensemble A n’as pas de maximum<br /> </p> @@ -865,16 +865,16 @@ L’ensemble A n’as pas de maximum<br /> </li> </ul> </div> -<div id="outline-container-org539689f" class="outline-4"> -<h4 id="org539689f">Ensemble B :</h4> -<div class="outline-text-4" id="text-org539689f"> +<div id="outline-container-org4c859a3" class="outline-4"> +<h4 id="org4c859a3">Ensemble B :</h4> +<div class="outline-text-4" id="text-org4c859a3"> <p> B = [-1 , 3[ ∩ ℚ<br /> </p> </div> <ul class="org-ul"> -<li><a id="org3a27e85"></a>Borne inférieure :<br /> -<div class="outline-text-5" id="text-org3a27e85"> +<li><a id="org5b309e4"></a>Borne inférieure :<br /> +<div class="outline-text-5" id="text-org5b309e4"> <p> Inf(B) = Max(inf([-1 , 3[) , inf(ℚ))<br /> </p> @@ -890,8 +890,8 @@ Puisse que ℚ n’as pas de Borne inférieure, donc par convention c’ </p> </div> </li> -<li><a id="org1097baa"></a>Borne supérieure :<br /> -<div class="outline-text-5" id="text-org1097baa"> +<li><a id="org1f4610f"></a>Borne supérieure :<br /> +<div class="outline-text-5" id="text-org1f4610f"> <p> Sup(B) = Min(sup([-1 ,3[) , sup(ℚ))<br /> </p> @@ -907,15 +907,15 @@ Puisse que ℚ n’as pas de Borne supérieure, donc par convention c’ </p> </div> </li> -<li><a id="org7a5b8da"></a>Minimum :<br /> -<div class="outline-text-5" id="text-org7a5b8da"> +<li><a id="orge42ed6f"></a>Minimum :<br /> +<div class="outline-text-5" id="text-orge42ed6f"> <p> <b>Min(B) = -1</b><br /> </p> </div> </li> -<li><a id="orgc128e50"></a>Maximum :<br /> -<div class="outline-text-5" id="text-orgc128e50"> +<li><a id="org6b202d0"></a>Maximum :<br /> +<div class="outline-text-5" id="text-org6b202d0"> <p> L’ensemble B n’as pas de Maximum<br /> </p> @@ -923,37 +923,37 @@ L’ensemble B n’as pas de Maximum<br /> </li> </ul> </div> -<div id="outline-container-orgced67f0" class="outline-4"> -<h4 id="orgced67f0">Ensemble C :</h4> -<div class="outline-text-4" id="text-orgced67f0"> +<div id="outline-container-org2ad9bb3" class="outline-4"> +<h4 id="org2ad9bb3">Ensemble C :</h4> +<div class="outline-text-4" id="text-org2ad9bb3"> <p> C = {3n ,n ∈ ℕ}<br /> </p> </div> <ul class="org-ul"> -<li><a id="org6f16567"></a>Borne inférieure :<br /> -<div class="outline-text-5" id="text-org6f16567"> +<li><a id="org78a462a"></a>Borne inférieure :<br /> +<div class="outline-text-5" id="text-org78a462a"> <p> Inf(C) = 0<br /> </p> </div> </li> -<li><a id="org6dbd187"></a>Borne supérieure :<br /> -<div class="outline-text-5" id="text-org6dbd187"> +<li><a id="orgd97c0b2"></a>Borne supérieure :<br /> +<div class="outline-text-5" id="text-orgd97c0b2"> <p> Sup(C) = +∞<br /> </p> </div> </li> -<li><a id="org70a65e5"></a>Minimum :<br /> -<div class="outline-text-5" id="text-org70a65e5"> +<li><a id="org86f58f9"></a>Minimum :<br /> +<div class="outline-text-5" id="text-org86f58f9"> <p> Min(C) = 0<br /> </p> </div> </li> -<li><a id="org74ee981"></a>Maximum :<br /> -<div class="outline-text-5" id="text-org74ee981"> +<li><a id="orgae16d77"></a>Maximum :<br /> +<div class="outline-text-5" id="text-orgae16d77"> <p> L’ensemble C n’as pas de Maximum<br /> </p> @@ -961,37 +961,37 @@ L’ensemble C n’as pas de Maximum<br /> </li> </ul> </div> -<div id="outline-container-orgedfa37a" class="outline-4"> -<h4 id="orgedfa37a">Ensemble D :</h4> -<div class="outline-text-4" id="text-orgedfa37a"> +<div id="outline-container-orgde49b00" class="outline-4"> +<h4 id="orgde49b00">Ensemble D :</h4> +<div class="outline-text-4" id="text-orgde49b00"> <p> D = {1 - 1/n , n ∈ ℕ*}<br /> </p> </div> <ul class="org-ul"> -<li><a id="org31ef9ff"></a>Borne inférieure :<br /> -<div class="outline-text-5" id="text-org31ef9ff"> +<li><a id="org820340a"></a>Borne inférieure :<br /> +<div class="outline-text-5" id="text-org820340a"> <p> Inf(D)= 0<br /> </p> </div> </li> -<li><a id="orga51cdcd"></a>Borne supérieure :<br /> -<div class="outline-text-5" id="text-orga51cdcd"> +<li><a id="org975f3e7"></a>Borne supérieure :<br /> +<div class="outline-text-5" id="text-org975f3e7"> <p> Sup(D)= 1<br /> </p> </div> </li> -<li><a id="org4a49ba3"></a>Minimum :<br /> -<div class="outline-text-5" id="text-org4a49ba3"> +<li><a id="org88d468a"></a>Minimum :<br /> +<div class="outline-text-5" id="text-org88d468a"> <p> Min(D)= 0<br /> </p> </div> </li> -<li><a id="org840590a"></a>Maximum :<br /> -<div class="outline-text-5" id="text-org840590a"> +<li><a id="org3aa5bd8"></a>Maximum :<br /> +<div class="outline-text-5" id="text-org3aa5bd8"> <p> L’ensemble D n’as pas de Maximum<br /> </p> @@ -999,9 +999,9 @@ L’ensemble D n’as pas de Maximum<br /> </li> </ul> </div> -<div id="outline-container-org13b9ce0" class="outline-4"> -<h4 id="org13b9ce0">Ensemble E :</h4> -<div class="outline-text-4" id="text-org13b9ce0"> +<div id="outline-container-orgb857fc5" class="outline-4"> +<h4 id="orgb857fc5">Ensemble E :</h4> +<div class="outline-text-4" id="text-orgb857fc5"> <p> E = { [2n + (-1)^n]/ n + 1 , n ∈ ℕ }<br /> </p> @@ -1022,8 +1022,8 @@ E = { [2n + (-1)^n]/ n + 1 , n ∈ ℕ }<br /> </p> </div> <ul class="org-ul"> -<li><a id="org526aea5"></a>Borne inférieure :<br /> -<div class="outline-text-5" id="text-org526aea5"> +<li><a id="org751c430"></a>Borne inférieure :<br /> +<div class="outline-text-5" id="text-org751c430"> <p> Inf(E) = Min(inf(F), inf(G))<br /> </p> @@ -1039,8 +1039,8 @@ Inf(F) = 1 ; Inf(G) = -1<br /> </p> </div> </li> -<li><a id="org64c7de8"></a>Borne supérieure :<br /> -<div class="outline-text-5" id="text-org64c7de8"> +<li><a id="orgc22974d"></a>Borne supérieure :<br /> +<div class="outline-text-5" id="text-orgc22974d"> <p> Sup(E) = Max(sup(F), sup(G))<br /> </p> @@ -1056,15 +1056,15 @@ sup(F) = +∞ ; sup(G) = +∞<br /> </p> </div> </li> -<li><a id="org2fc9819"></a>Minimum :<br /> -<div class="outline-text-5" id="text-org2fc9819"> +<li><a id="orga73b811"></a>Minimum :<br /> +<div class="outline-text-5" id="text-orga73b811"> <p> Min(E)= -1<br /> </p> </div> </li> -<li><a id="org34b0670"></a>Maximum :<br /> -<div class="outline-text-5" id="text-org34b0670"> +<li><a id="org1685ba6"></a>Maximum :<br /> +<div class="outline-text-5" id="text-org1685ba6"> <p> E n’as pas de maximum<br /> </p> @@ -1073,20 +1073,20 @@ E n’as pas de maximum<br /> </ul> </div> </div> -<div id="outline-container-org227cd11" class="outline-3"> -<h3 id="org227cd11">Exo 2 :</h3> -<div class="outline-text-3" id="text-org227cd11"> +<div id="outline-container-org3241c28" class="outline-3"> +<h3 id="org3241c28">Exo 2 :</h3> +<div class="outline-text-3" id="text-org3241c28"> </div> -<div id="outline-container-org1c6a24b" class="outline-4"> -<h4 id="org1c6a24b">Ensemble A :</h4> -<div class="outline-text-4" id="text-org1c6a24b"> +<div id="outline-container-org77ffa3e" class="outline-4"> +<h4 id="org77ffa3e">Ensemble A :</h4> +<div class="outline-text-4" id="text-org77ffa3e"> <p> A = {x ∈ ℝ , 0 < x <√3}<br /> </p> </div> <ul class="org-ul"> -<li><a id="orga3775a6"></a>Borné<br /> -<div class="outline-text-5" id="text-orga3775a6"> +<li><a id="org3bdaec9"></a>Borné<br /> +<div class="outline-text-5" id="text-org3bdaec9"> <p> <b>Oui</b>, Inf(A)= 0 ; Sup(A)=√3<br /> </p> @@ -1094,16 +1094,16 @@ A = {x ∈ ℝ , 0 < x <√3}<br /> </li> </ul> </div> -<div id="outline-container-org51c6bfc" class="outline-4"> -<h4 id="org51c6bfc">Ensemble B :</h4> -<div class="outline-text-4" id="text-org51c6bfc"> +<div id="outline-container-org151d601" class="outline-4"> +<h4 id="org151d601">Ensemble B :</h4> +<div class="outline-text-4" id="text-org151d601"> <p> B = { x ∈ ℝ , 1/2 < sin x <√3/2} ;<br /> </p> </div> <ul class="org-ul"> -<li><a id="orgba6c8c3"></a>Borné<br /> -<div class="outline-text-5" id="text-orgba6c8c3"> +<li><a id="orgf630bc2"></a>Borné<br /> +<div class="outline-text-5" id="text-orgf630bc2"> <p> <b>∀ x ∈ B, sin x > 1/2 ∴ Inf(B)= 1/2</b><br /> </p> @@ -1116,16 +1116,16 @@ B = { x ∈ ℝ , 1/2 < sin x <√3/2} ;<br /> </li> </ul> </div> -<div id="outline-container-org7accf6a" class="outline-4"> -<h4 id="org7accf6a">Ensemble C :</h4> -<div class="outline-text-4" id="text-org7accf6a"> +<div id="outline-container-orgbc1efd9" class="outline-4"> +<h4 id="orgbc1efd9">Ensemble C :</h4> +<div class="outline-text-4" id="text-orgbc1efd9"> <p> C = {x ∈ ℝ , x³ > 3}<br /> </p> </div> <ul class="org-ul"> -<li><a id="orgf7f441a"></a>Minoré<br /> -<div class="outline-text-5" id="text-orgf7f441a"> +<li><a id="orga289bfe"></a>Minoré<br /> +<div class="outline-text-5" id="text-orga289bfe"> <p> <b>∀ x ∈ C, x³ > 3 ∴ Inf(C)= 3</b><br /> </p> @@ -1133,16 +1133,16 @@ C = {x ∈ ℝ , x³ > 3}<br /> </li> </ul> </div> -<div id="outline-container-org4f20a91" class="outline-4"> -<h4 id="org4f20a91">Ensemble D :</h4> -<div class="outline-text-4" id="text-org4f20a91"> +<div id="outline-container-org0eda8d2" class="outline-4"> +<h4 id="org0eda8d2">Ensemble D :</h4> +<div class="outline-text-4" id="text-org0eda8d2"> <p> D = {x ∈ ℝ , e^x < 1/2}<br /> </p> </div> <ul class="org-ul"> -<li><a id="orgf68d2c2"></a>Borné<br /> -<div class="outline-text-5" id="text-orgf68d2c2"> +<li><a id="orgeb91bff"></a>Borné<br /> +<div class="outline-text-5" id="text-orgeb91bff"> <p> <b>∀ x ∈ C, e^x > 0 ∴ Inf(C)= 0</b><br /> </p> @@ -1155,16 +1155,16 @@ D = {x ∈ ℝ , e^x < 1/2}<br /> </li> </ul> </div> -<div id="outline-container-org9a20270" class="outline-4"> -<h4 id="org9a20270">Ensemble E :</h4> -<div class="outline-text-4" id="text-org9a20270"> +<div id="outline-container-org9b9b691" class="outline-4"> +<h4 id="org9b9b691">Ensemble E :</h4> +<div class="outline-text-4" id="text-org9b9b691"> <p> E = {x ∈ ℝ , ∃ p ∈ ℕ* : x = √2/p}<br /> </p> </div> <ul class="org-ul"> -<li><a id="org1e6a8bc"></a>Majoré<br /> -<div class="outline-text-5" id="text-org1e6a8bc"> +<li><a id="org5f1feca"></a>Majoré<br /> +<div class="outline-text-5" id="text-org5f1feca"> <p> p = √2/x . Donc : <b>Sup(E)=1</b><br /> </p> @@ -1173,16 +1173,16 @@ p = √2/x . Donc : <b>Sup(E)=1</b><br /> </ul> </div> </div> -<div id="outline-container-orgae3b875" class="outline-3"> -<h3 id="orgae3b875">Exo 3 :</h3> -<div class="outline-text-3" id="text-orgae3b875"> +<div id="outline-container-org36dc1da" class="outline-3"> +<h3 id="org36dc1da">Exo 3 :</h3> +<div class="outline-text-3" id="text-org36dc1da"> <p> U0 = 3/2 ; U(n+1) = (Un - 1)² + 1<br /> </p> </div> -<div id="outline-container-org261f974" class="outline-4"> -<h4 id="org261f974">Question 1 :</h4> -<div class="outline-text-4" id="text-org261f974"> +<div id="outline-container-org7999092" class="outline-4"> +<h4 id="org7999092">Question 1 :</h4> +<div class="outline-text-4" id="text-org7999092"> <p> Montrer que : ∀ n ∈ ℕ , 1 < Un < 2 .<br /> </p> @@ -1198,8 +1198,8 @@ Montrer que : ∀ n ∈ ℕ , 1 < Un < 2 .<br /> </p> </div> <ul class="org-ul"> -<li><a id="org59c26c1"></a>Raisonnement par récurrence :<br /> -<div class="outline-text-5" id="text-org59c26c1"> +<li><a id="orgd5b9f21"></a>Raisonnement par récurrence :<br /> +<div class="outline-text-5" id="text-orgd5b9f21"> <p> P(n) : ∀ n ∈ ℕ ; 1 < Un < 2<br /> </p> @@ -1222,9 +1222,9 @@ On suppose que P(n) est vraie et on vérifie P(n+1) pour une contradiction<br /> </li> </ul> </div> -<div id="outline-container-orge6e1ce3" class="outline-4"> -<h4 id="orge6e1ce3">Question 2 :</h4> -<div class="outline-text-4" id="text-orge6e1ce3"> +<div id="outline-container-orgb9f7a15" class="outline-4"> +<h4 id="orgb9f7a15">Question 2 :</h4> +<div class="outline-text-4" id="text-orgb9f7a15"> <p> Montrer que (Un)n est strictement monotone :<br /> </p> @@ -1247,20 +1247,20 @@ On déduit que <b>Un² - 3Un + 2</b> est négatif sur [1 , 2] et positif en deho </div> </div> </div> -<div id="outline-container-orge45c9b7" class="outline-2"> -<h2 id="orge45c9b7">4th cours (Suite) : <i>Oct 10</i></h2> -<div class="outline-text-2" id="text-orge45c9b7"> +<div id="outline-container-org3da135e" class="outline-2"> +<h2 id="org3da135e">4th cours (Suite) : <i>Oct 10</i></h2> +<div class="outline-text-2" id="text-org3da135e"> </div> -<div id="outline-container-org58b09d5" class="outline-3"> -<h3 id="org58b09d5">Les suites convergentes</h3> -<div class="outline-text-3" id="text-org58b09d5"> +<div id="outline-container-org639877a" class="outline-3"> +<h3 id="org639877a">Les suites convergentes</h3> +<div class="outline-text-3" id="text-org639877a"> <p> Soit (Un)n est une suite convergente si lim Un n–> +∞ = l<br /> </p> </div> -<div id="outline-container-org820f474" class="outline-4"> -<h4 id="org820f474">Remarque :</h4> -<div class="outline-text-4" id="text-org820f474"> +<div id="outline-container-orgce5e8f7" class="outline-4"> +<h4 id="orgce5e8f7">Remarque :</h4> +<div class="outline-text-4" id="text-orgce5e8f7"> <ol class="org-ol"> <li>Un est une suite convergente alors Un est bornee<br /></li> <li>Un est une suite convergente lim Un n—> +∞ = l ⇔ lim |Un| n—> +∞ = |l|<br /></li> @@ -1277,32 +1277,32 @@ Soit (Un)n est une suite convergente si lim Un n–> +∞ = l<br /> </div> </div> </div> -<div id="outline-container-orge043bff" class="outline-3"> -<h3 id="orge043bff">Theoreme d’encadrement</h3> -<div class="outline-text-3" id="text-orge043bff"> +<div id="outline-container-orga659f1f" class="outline-3"> +<h3 id="orga659f1f">Theoreme d’encadrement</h3> +<div class="outline-text-3" id="text-orga659f1f"> <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<br /> </p> </div> </div> -<div id="outline-container-org8460bc2" class="outline-3"> -<h3 id="org8460bc2">Suites arithmetiques</h3> -<div class="outline-text-3" id="text-org8460bc2"> +<div id="outline-container-org4c1ed41" class="outline-3"> +<h3 id="org4c1ed41">Suites arithmetiques</h3> +<div class="outline-text-3" id="text-org4c1ed41"> <p> Un est une suite arithmetique si : U(n+1) = Un + r ; r etant la raison de la suite<br /> </p> </div> -<div id="outline-container-org51b313a" class="outline-4"> -<h4 id="org51b313a">Forme general</h4> -<div class="outline-text-4" id="text-org51b313a"> +<div id="outline-container-org5b887fd" class="outline-4"> +<h4 id="org5b887fd">Forme general</h4> +<div class="outline-text-4" id="text-org5b887fd"> <p> <b>Un = U0 + nr</b> ; <b>Un = Up + (n - p)r</b><br /> </p> </div> </div> -<div id="outline-container-org63ac12f" class="outline-4"> -<h4 id="org63ac12f">Somme des n premiers termes</h4> -<div class="outline-text-4" id="text-org63ac12f"> +<div id="outline-container-orgbd36410" class="outline-4"> +<h4 id="orgbd36410">Somme des n premiers termes</h4> +<div class="outline-text-4" id="text-orgbd36410"> <p> Un est une suite arithmetique, Sn = [(U0 + Un)(n + 1)]/2<br /> </p> @@ -1314,21 +1314,21 @@ Sn = (n, k = 0)ΣUk est une somme partielle et lim Sn n->+∞ = k≥0ΣUk est </div> </div> </div> -<div id="outline-container-org24bc205" class="outline-3"> -<h3 id="org24bc205">Suites géométriques</h3> -<div class="outline-text-3" id="text-org24bc205"> +<div id="outline-container-orge060a6b" class="outline-3"> +<h3 id="orge060a6b">Suites géométriques</h3> +<div class="outline-text-3" id="text-orge060a6b"> </div> -<div id="outline-container-orgc70d4f0" class="outline-4"> -<h4 id="orgc70d4f0">Forme general</h4> -<div class="outline-text-4" id="text-orgc70d4f0"> +<div id="outline-container-org7eb64b7" class="outline-4"> +<h4 id="org7eb64b7">Forme general</h4> +<div class="outline-text-4" id="text-org7eb64b7"> <p> <b>Un = U0 x r^n</b><br /> </p> </div> </div> -<div id="outline-container-org276a52e" class="outline-4"> -<h4 id="org276a52e">Somme des n premiers termes</h4> -<div class="outline-text-4" id="text-org276a52e"> +<div id="outline-container-org4a1c78c" class="outline-4"> +<h4 id="org4a1c78c">Somme des n premiers termes</h4> +<div class="outline-text-4" id="text-org4a1c78c"> <p> n ∈ ℕ\{1} Sn = U0 (1 - r^(n+1))/1-r<br /> </p> @@ -1336,13 +1336,13 @@ n ∈ ℕ\{1} Sn = U0 (1 - r^(n+1))/1-r<br /> </div> </div> </div> -<div id="outline-container-orgb4ceb77" class="outline-2"> -<h2 id="orgb4ceb77">5th cours (suite) : <i>Oct 12</i></h2> -<div class="outline-text-2" id="text-orgb4ceb77"> +<div id="outline-container-org9ad98cf" class="outline-2"> +<h2 id="org9ad98cf">5th cours (suite) : <i>Oct 12</i></h2> +<div class="outline-text-2" id="text-org9ad98cf"> </div> -<div id="outline-container-org5f6fa60" class="outline-3"> -<h3 id="org5f6fa60">Suites adjacentes:</h3> -<div class="outline-text-3" id="text-org5f6fa60"> +<div id="outline-container-org3ef59f2" class="outline-3"> +<h3 id="org3ef59f2">Suites adjacentes:</h3> +<div class="outline-text-3" id="text-org3ef59f2"> <p> Soient (Un) et (Vn) deux suites, elles sont adjacentes si:<br /> </p> @@ -1353,16 +1353,16 @@ Soient (Un) et (Vn) deux suites, elles sont adjacentes si:<br /> </ol> </div> </div> -<div id="outline-container-orgea8a031" class="outline-3"> -<h3 id="orgea8a031">Suites extraites (sous-suites):</h3> -<div class="outline-text-3" id="text-orgea8a031"> +<div id="outline-container-org05716a0" class="outline-3"> +<h3 id="org05716a0">Suites extraites (sous-suites):</h3> +<div class="outline-text-3" id="text-org05716a0"> <p> Soit (Un) une suite: ;U: ℕ -—> ℝ ; n -—> Un ;ϕ: ℕ -—> ℕ ; n -—> ϕn ;(U(ϕ(n))) est appelée une sous suite de (Un) ou bien une suite extraite.<br /> </p> </div> -<div id="outline-container-org0a4213f" class="outline-4"> -<h4 id="org0a4213f">Remarques:</h4> -<div class="outline-text-4" id="text-org0a4213f"> +<div id="outline-container-org312cfda" class="outline-4"> +<h4 id="org312cfda">Remarques:</h4> +<div class="outline-text-4" id="text-org312cfda"> <ol class="org-ol"> <li>Si (Un) converge ⇒ ∀ n ∈ ℕ , U(ϕ(n)) converge aussi.<br /></li> <li>Mais le contraire n’es pas toujours vrais.<br /></li> @@ -1371,25 +1371,25 @@ Soit (Un) une suite: ;U: ℕ -—> ℝ ; n -—> Un ;ϕ: ℕ - </div> </div> </div> -<div id="outline-container-orgec23ceb" class="outline-3"> -<h3 id="orgec23ceb">Suites de Cauchy:</h3> -<div class="outline-text-3" id="text-orgec23ceb"> +<div id="outline-container-orgbfa31ac" class="outline-3"> +<h3 id="orgbfa31ac">Suites de Cauchy:</h3> +<div class="outline-text-3" id="text-orgbfa31ac"> <p> (Un) n ∈ ℕ est une suite de Cauchy Si ; ;∀ ε > 0 , ∃ N ∈ ℕ ; ∀ n > m > N ; |Un - Um| < ε<br /> </p> </div> -<div id="outline-container-org04bdc9a" class="outline-4"> -<h4 id="org04bdc9a">Remarque :</h4> -<div class="outline-text-4" id="text-org04bdc9a"> +<div id="outline-container-org60c9452" class="outline-4"> +<h4 id="org60c9452">Remarque :</h4> +<div class="outline-text-4" id="text-org60c9452"> <ol class="org-ol"> <li>Toute suite convergente est une suite de Cauchy et toute suite Cauchy est une suite convergente<br /></li> </ol> </div> </div> </div> -<div id="outline-container-orgc639b18" class="outline-3"> -<h3 id="orgc639b18">Théorème de Bolzano Weirstrass:</h3> -<div class="outline-text-3" id="text-orgc639b18"> +<div id="outline-container-org678d2ef" class="outline-3"> +<h3 id="org678d2ef">Théorème de Bolzano Weirstrass:</h3> +<div class="outline-text-3" id="text-org678d2ef"> <p> On peut extraire une sous suite convergente de toute suite bornée<br /> </p> @@ -1399,7 +1399,7 @@ On peut extraire une sous suite convergente de toute suite bornée<br /> </div> <div id="postamble" class="status"> <p class="author">Author: Crystal</p> -<p class="date">Created: 2023-11-01 Wed 20:10</p> +<p class="date">Created: 2023-11-01 Wed 20:16</p> </div> </body> </html> \ No newline at end of file diff --git a/uni_notes/architecture.html b/uni_notes/architecture.html index cfdb462..290e923 100755 --- a/uni_notes/architecture.html +++ b/uni_notes/architecture.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <head> -<!-- 2023-11-01 Wed 20:10 --> +<!-- 2023-11-01 Wed 20:16 --> <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> @@ -11,7 +11,7 @@ <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"/> -<link rel="icon" type="image/x-icon" href="https://crystal.tilde.institute/favicon.ico"> +<link rel="icon" type="image/x-icon" href="https://crystal.tilde.institute/favicon.png"> </head> <body> <div id="org-div-home-and-up"> @@ -24,59 +24,59 @@ <h2>Table of Contents</h2> <div id="text-table-of-contents" role="doc-toc"> <ul> -<li><a href="#org2d531e4">Premier cours : Les systémes de numération <i>Sep 27</i> :</a> +<li><a href="#org3fa8932">Premier cours : Les systémes de numération <i>Sep 27</i> :</a> <ul> <li> <ul> -<li><a href="#orgf46fb48"><b>Examples :</b></a></li> +<li><a href="#orgb23a1f3"><b>Examples :</b></a></li> </ul> </li> -<li><a href="#org7dcbcb0">Comment passer d’un systéme a base 10 a un autre</a> +<li><a href="#orga02a1f6">Comment passer d’un systéme a base 10 a un autre</a> <ul> -<li><a href="#org82fa425">Pour les chiffres entiers :</a></li> -<li><a href="#org2f77f80">Pour les chiffres non entiers :</a></li> +<li><a href="#org8c7a5f5">Pour les chiffres entiers :</a></li> +<li><a href="#org6378ac0">Pour les chiffres non entiers :</a></li> </ul> </li> </ul> </li> -<li><a href="#org0cd9bb6">2nd cours : Les systèmes de numération (Suite) <i>Oct 3</i> :</a> +<li><a href="#org251a561">2nd cours : Les systèmes de numération (Suite) <i>Oct 3</i> :</a> <ul> -<li><a href="#org4cf9200">Comment passer d’une base N a la base 10 :</a></li> -<li><a href="#org6163b5c">Comment passer d’une base N a une base N^(n) :</a> +<li><a href="#org2a38085">Comment passer d’une base N a la base 10 :</a></li> +<li><a href="#orgc0fdca1">Comment passer d’une base N a une base N^(n) :</a> <ul> -<li><a href="#orgcc06569">Exemple :</a></li> +<li><a href="#orgba624be">Exemple :</a></li> </ul> </li> -<li><a href="#org71b978d">L’arithmétique binaire :</a> +<li><a href="#org789d800">L’arithmétique binaire :</a> <ul> -<li><a href="#org51f73ce">L’addition :</a></li> -<li><a href="#orgb36f4bd">La soustraction :</a></li> +<li><a href="#orgac84614">L’addition :</a></li> +<li><a href="#org4829f28">La soustraction :</a></li> </ul> </li> -<li><a href="#orgdae30cd">TP N°1 :</a> +<li><a href="#org110b8dd">TP N°1 :</a> <ul> -<li><a href="#orgf474d31">Exo1:</a></li> -<li><a href="#org33da6ab">Exo2:</a></li> -<li><a href="#org401e1ed">Exo3:</a></li> +<li><a href="#org54d7fdf">Exo1:</a></li> +<li><a href="#org6654eb2">Exo2:</a></li> +<li><a href="#org9da39d0">Exo3:</a></li> </ul> </li> -<li><a href="#org68f9820">L’arithmétique binaire (Suite): <i>Oct 4</i></a> +<li><a href="#org2039fb1">L’arithmétique binaire (Suite): <i>Oct 4</i></a> <ul> -<li><a href="#org69a2f51">La multiplication :</a></li> -<li><a href="#org62ef065">La division :</a></li> +<li><a href="#orgfc45d53">La multiplication :</a></li> +<li><a href="#org91fcccc">La division :</a></li> </ul> </li> </ul> </li> -<li><a href="#org12d79ca">4th cours : Le codage <i>Oct 10</i></a> +<li><a href="#orgcbf8da9">4th cours : Le codage <i>Oct 10</i></a> <ul> -<li><a href="#orgd5b1107">Le codage des entiers positifs</a></li> -<li><a href="#orgbc351c0">Le codage des nombres relatifs</a> +<li><a href="#org34693c1">Le codage des entiers positifs</a></li> +<li><a href="#org3c8ed5c">Le codage des nombres relatifs</a> <ul> -<li><a href="#org2110907">Remarque</a></li> -<li><a href="#org3c534f2">Le codage en signe + valeur absolue (SVA):</a></li> -<li><a href="#org0259b15">Codage en compliment a 1 (CR):</a></li> -<li><a href="#orgf83cdec">Codage en compliment a 2 (CV):</a></li> +<li><a href="#orgb2d4951">Remarque</a></li> +<li><a href="#orgca1d761">Le codage en signe + valeur absolue (SVA):</a></li> +<li><a href="#orgd2c678f">Codage en compliment a 1 (CR):</a></li> +<li><a href="#org9961620">Codage en compliment a 2 (CV):</a></li> </ul> </li> </ul> @@ -84,9 +84,9 @@ </ul> </div> </div> -<div id="outline-container-org2d531e4" class="outline-2"> -<h2 id="org2d531e4">Premier cours : Les systémes de numération <i>Sep 27</i> :</h2> -<div class="outline-text-2" id="text-org2d531e4"> +<div id="outline-container-org3fa8932" class="outline-2"> +<h2 id="org3fa8932">Premier cours : Les systémes de numération <i>Sep 27</i> :</h2> +<div class="outline-text-2" id="text-org3fa8932"> <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:<br /> </p> @@ -97,9 +97,9 @@ Un système de numération est une méthode pour représenter des nombres à l&r <li>Des régles de représentations des nombres<br /></li> </ol> </div> -<div id="outline-container-orgf46fb48" class="outline-4"> -<h4 id="orgf46fb48"><b>Examples :</b></h4> -<div class="outline-text-4" id="text-orgf46fb48"> +<div id="outline-container-orgb23a1f3" class="outline-4"> +<h4 id="orgb23a1f3"><b>Examples :</b></h4> +<div class="outline-text-4" id="text-orgb23a1f3"> <p> <i>B10 est un systéme de numération caractérisé par:</i><br /> </p> @@ -127,16 +127,16 @@ A : 10 ; B : 11 ; C : 12 ; D : 13 ; E : 14 ; F : 15<br /> </ul> </div> </div> -<div id="outline-container-org7dcbcb0" class="outline-3"> -<h3 id="org7dcbcb0">Comment passer d’un systéme a base 10 a un autre</h3> -<div class="outline-text-3" id="text-org7dcbcb0"> +<div id="outline-container-orga02a1f6" class="outline-3"> +<h3 id="orga02a1f6">Comment passer d’un systéme a base 10 a un autre</h3> +<div class="outline-text-3" id="text-orga02a1f6"> <p> On symbolise un chiffre dans la base x par : (Nombre)x<br /> </p> </div> -<div id="outline-container-org82fa425" class="outline-4"> -<h4 id="org82fa425">Pour les chiffres entiers :</h4> -<div class="outline-text-4" id="text-org82fa425"> +<div id="outline-container-org8c7a5f5" class="outline-4"> +<h4 id="org8c7a5f5">Pour les chiffres entiers :</h4> +<div class="outline-text-4" id="text-org8c7a5f5"> <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><br /> </p> @@ -164,8 +164,8 @@ On dévise 3257 par 16, et les restants de la division serra la valeur en base16 </p> </div> <ul class="org-ul"> -<li><a id="orgf2707e8"></a><b>Conclusion:</b><br /> -<div class="outline-text-5" id="text-orgf2707e8"> +<li><a id="org9a466ee"></a><b>Conclusion:</b><br /> +<div class="outline-text-5" id="text-org9a466ee"> <p> (3257)10 -—> (CB9)16<br /> </p> @@ -173,9 +173,9 @@ On dévise 3257 par 16, et les restants de la division serra la valeur en base16 </li> </ul> </div> -<div id="outline-container-org2f77f80" class="outline-4"> -<h4 id="org2f77f80">Pour les chiffres non entiers :</h4> -<div class="outline-text-4" id="text-org2f77f80"> +<div id="outline-container-org6378ac0" class="outline-4"> +<h4 id="org6378ac0">Pour les chiffres non entiers :</h4> +<div class="outline-text-4" id="text-org6378ac0"> <p> <b>On fait la division successive pour la partie entiére, et une multiplication successive pour la partie rationelle:</b><br /> </p> @@ -228,13 +228,13 @@ On a déja la partie entiére donc on s’occupe de la partie aprés la virg </div> </div> </div> -<div id="outline-container-org0cd9bb6" class="outline-2"> -<h2 id="org0cd9bb6">2nd cours : Les systèmes de numération (Suite) <i>Oct 3</i> :</h2> -<div class="outline-text-2" id="text-org0cd9bb6"> +<div id="outline-container-org251a561" class="outline-2"> +<h2 id="org251a561">2nd cours : Les systèmes de numération (Suite) <i>Oct 3</i> :</h2> +<div class="outline-text-2" id="text-org251a561"> </div> -<div id="outline-container-org4cf9200" class="outline-3"> -<h3 id="org4cf9200">Comment passer d’une base N a la base 10 :</h3> -<div class="outline-text-3" id="text-org4cf9200"> +<div id="outline-container-org2a38085" class="outline-3"> +<h3 id="org2a38085">Comment passer d’une base N a la base 10 :</h3> +<div class="outline-text-3" id="text-org2a38085"> <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 :<br /> </p> @@ -255,16 +255,16 @@ Et pour passer a la base 10, il suffit d’appliquer cette formule : <b>Chif </p> </div> </div> -<div id="outline-container-org6163b5c" class="outline-3"> -<h3 id="org6163b5c">Comment passer d’une base N a une base N^(n) :</h3> -<div class="outline-text-3" id="text-org6163b5c"> +<div id="outline-container-orgc0fdca1" class="outline-3"> +<h3 id="orgc0fdca1">Comment passer d’une base N a une base N^(n) :</h3> +<div class="outline-text-3" id="text-orgc0fdca1"> <p> Si il ya une relation entre une base et une autre, on peut directement transformer vers cette base.<br /> </p> </div> -<div id="outline-container-orgcc06569" class="outline-4"> -<h4 id="orgcc06569">Exemple :</h4> -<div class="outline-text-4" id="text-orgcc06569"> +<div id="outline-container-orgba624be" class="outline-4"> +<h4 id="orgba624be">Exemple :</h4> +<div class="outline-text-4" id="text-orgba624be"> <p> Pour passer de la base 2 a la base 8 (8 qui est 2³) on découpe les chiffres 3 par 3<br /> </p> @@ -376,13 +376,13 @@ Maintenant il suffit de trouver l’équivalent de la base2 en base8 :<br /> </div> </div> </div> -<div id="outline-container-org71b978d" class="outline-3"> -<h3 id="org71b978d">L’arithmétique binaire :</h3> -<div class="outline-text-3" id="text-org71b978d"> +<div id="outline-container-org789d800" class="outline-3"> +<h3 id="org789d800">L’arithmétique binaire :</h3> +<div class="outline-text-3" id="text-org789d800"> </div> -<div id="outline-container-org51f73ce" class="outline-4"> -<h4 id="org51f73ce">L’addition :</h4> -<div class="outline-text-4" id="text-org51f73ce"> +<div id="outline-container-orgac84614" class="outline-4"> +<h4 id="orgac84614">L’addition :</h4> +<div class="outline-text-4" id="text-orgac84614"> <p> 0 + 0 = 0 On retiens 0<br /> </p> @@ -413,9 +413,9 @@ Donc 0110 + 1101 = 10011<br /> </p> </div> </div> -<div id="outline-container-orgb36f4bd" class="outline-4"> -<h4 id="orgb36f4bd">La soustraction :</h4> -<div class="outline-text-4" id="text-orgb36f4bd"> +<div id="outline-container-org4829f28" class="outline-4"> +<h4 id="org4829f28">La soustraction :</h4> +<div class="outline-text-4" id="text-org4829f28"> <p> 0 - 0 = 0 On emprunt = 0<br /> </p> @@ -437,13 +437,13 @@ Donc 0110 + 1101 = 10011<br /> </div> </div> </div> -<div id="outline-container-orgdae30cd" class="outline-3"> -<h3 id="orgdae30cd">TP N°1 :</h3> -<div class="outline-text-3" id="text-orgdae30cd"> +<div id="outline-container-org110b8dd" class="outline-3"> +<h3 id="org110b8dd">TP N°1 :</h3> +<div class="outline-text-3" id="text-org110b8dd"> </div> -<div id="outline-container-orgf474d31" class="outline-4"> -<h4 id="orgf474d31">Exo1:</h4> -<div class="outline-text-4" id="text-orgf474d31"> +<div id="outline-container-org54d7fdf" class="outline-4"> +<h4 id="org54d7fdf">Exo1:</h4> +<div class="outline-text-4" id="text-org54d7fdf"> <table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> @@ -511,15 +511,15 @@ Donc 0110 + 1101 = 10011<br /> </table> </div> <ul class="org-ul"> -<li><a id="org49d5997"></a>(10110,11)2<br /> -<div class="outline-text-5" id="text-org49d5997"> +<li><a id="orga7c42ee"></a>(10110,11)2<br /> +<div class="outline-text-5" id="text-orga7c42ee"> <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<br /> </p> </div> <ul class="org-ul"> -<li><a id="orgad57016"></a>(22,75)10 -—> (3)<br /> -<div class="outline-text-6" id="text-orgad57016"> +<li><a id="org02d1304"></a>(22,75)10 -—> (3)<br /> +<div class="outline-text-6" id="text-org02d1304"> <p> 22/3 = 7 R <b>1</b> ; 7/3 = 2 R <b>1</b> ; 2/3 = 0 R <b>2</b><br /> </p> @@ -535,8 +535,8 @@ Donc 0110 + 1101 = 10011<br /> </p> </div> </li> -<li><a id="org2c62f57"></a>(10110,11)2 -—> (8)<br /> -<div class="outline-text-6" id="text-org2c62f57"> +<li><a id="org00a17b7"></a>(10110,11)2 -—> (8)<br /> +<div class="outline-text-6" id="text-org00a17b7"> <p> 8 = 2³ ; (010 110,110)2 -—> (?)8<br /> </p> @@ -557,8 +557,8 @@ En utilisant le tableau 3bits :<br /> </p> </div> </li> -<li><a id="org9088b8f"></a>(22,75)10 -—> (16)<br /> -<div class="outline-text-6" id="text-org9088b8f"> +<li><a id="org8bffc7b"></a>(22,75)10 -—> (16)<br /> +<div class="outline-text-6" id="text-org8bffc7b"> <p> 22/16 = 1 R <b>6</b> ; 1/16 : 0 R <b>F</b><br /> </p> @@ -576,15 +576,15 @@ En utilisant le tableau 3bits :<br /> </li> </ul> </li> -<li><a id="org6a53f73"></a>(1254,1)8<br /> -<div class="outline-text-5" id="text-org6a53f73"> +<li><a id="orgd6d1187"></a>(1254,1)8<br /> +<div class="outline-text-5" id="text-orgd6d1187"> <p> 4 x 8° + 5 x 8¹ + 2 x 8² + 1 x 8³ + 1 x 8¯¹ = (684,125)10<br /> </p> </div> <ul class="org-ul"> -<li><a id="orgd523ee6"></a>(1254,1)8 -—> (?)2<br /> -<div class="outline-text-6" id="text-orgd523ee6"> +<li><a id="orgfe79941"></a>(1254,1)8 -—> (?)2<br /> +<div class="outline-text-6" id="text-orgfe79941"> <p> En utilisant le tableau 3bits :<br /> </p> @@ -600,8 +600,8 @@ En utilisant le tableau 3bits :<br /> </p> </div> </li> -<li><a id="org6d138b3"></a>(684,125)10 -—> (?)3<br /> -<div class="outline-text-6" id="text-org6d138b3"> +<li><a id="org5468c02"></a>(684,125)10 -—> (?)3<br /> +<div class="outline-text-6" id="text-org5468c02"> <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><br /> </p> @@ -617,8 +617,8 @@ En utilisant le tableau 3bits :<br /> </p> </div> </li> -<li><a id="org215cb04"></a>(684,125)10 -—> (?)16<br /> -<div class="outline-text-6" id="text-org215cb04"> +<li><a id="org8acf355"></a>(684,125)10 -—> (?)16<br /> +<div class="outline-text-6" id="text-org8acf355"> <p> 684/16 = 42 R <b>C</b> ; 42/16 = 2 R <b>A</b> ; 2/16 0 R <b>2</b><br /> </p> @@ -636,15 +636,15 @@ En utilisant le tableau 3bits :<br /> </li> </ul> </li> -<li><a id="orgd965e55"></a>(F5B,A)16<br /> -<div class="outline-text-5" id="text-orgd965e55"> +<li><a id="orgaed5ea2"></a>(F5B,A)16<br /> +<div class="outline-text-5" id="text-orgaed5ea2"> <p> 11 x 16° + 5 x 16 + 15 x 16² + 10 x 16¯¹ = (3931,625)10<br /> </p> </div> <ul class="org-ul"> -<li><a id="org9525d94"></a>(3931,625)10 -—> (8)<br /> -<div class="outline-text-6" id="text-org9525d94"> +<li><a id="org56c0052"></a>(3931,625)10 -—> (8)<br /> +<div class="outline-text-6" id="text-org56c0052"> <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><br /> </p> @@ -660,8 +660,8 @@ En utilisant le tableau 3bits :<br /> </p> </div> </li> -<li><a id="org7bbb1fc"></a>(7533,5)8 -—> (2)<br /> -<div class="outline-text-6" id="text-org7bbb1fc"> +<li><a id="org64e9962"></a>(7533,5)8 -—> (2)<br /> +<div class="outline-text-6" id="text-org64e9962"> <p> En utilisant le tableau 3bits<br /> </p> @@ -671,8 +671,8 @@ En utilisant le tableau 3bits<br /> </p> </div> </li> -<li><a id="orgf841adc"></a>(3931,625)10 -—> (3)<br /> -<div class="outline-text-6" id="text-orgf841adc"> +<li><a id="org2850a22"></a>(3931,625)10 -—> (3)<br /> +<div class="outline-text-6" id="text-org2850a22"> <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><br /> </p> @@ -690,8 +690,8 @@ En utilisant le tableau 3bits<br /> </li> </ul> </li> -<li><a id="orgef76490"></a>(52,38)10<br /> -<div class="outline-text-5" id="text-orgef76490"> +<li><a id="org2ee5d93"></a>(52,38)10<br /> +<div class="outline-text-5" id="text-org2ee5d93"> <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><br /> </p> @@ -707,8 +707,8 @@ En utilisant le tableau 3bits<br /> </p> </div> <ul class="org-ul"> -<li><a id="org40e5ef4"></a>(52,38)10 -—> (3)<br /> -<div class="outline-text-6" id="text-org40e5ef4"> +<li><a id="org3f89d6a"></a>(52,38)10 -—> (3)<br /> +<div class="outline-text-6" id="text-org3f89d6a"> <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><br /> </p> @@ -724,8 +724,8 @@ En utilisant le tableau 3bits<br /> </p> </div> </li> -<li><a id="org6c9dade"></a>(110100,011)2 -—> (8)<br /> -<div class="outline-text-6" id="text-org6c9dade"> +<li><a id="org388e3c0"></a>(110100,011)2 -—> (8)<br /> +<div class="outline-text-6" id="text-org388e3c0"> <p> En utilisant le tableau 3bits:<br /> </p> @@ -736,8 +736,8 @@ En utilisant le tableau 3bits:<br /> </p> </div> </li> -<li><a id="orgef7c31b"></a>(52,38)10 -—> (16)<br /> -<div class="outline-text-6" id="text-orgef7c31b"> +<li><a id="orgb909e73"></a>(52,38)10 -—> (16)<br /> +<div class="outline-text-6" id="text-orgb909e73"> <p> 52/16 = 3 R <b>4</b> ; 3/16 = 0 R <b>3</b><br /> </p> @@ -755,15 +755,15 @@ En utilisant le tableau 3bits:<br /> </li> </ul> </li> -<li><a id="org4fd998e"></a>(23,5)3<br /> -<div class="outline-text-5" id="text-org4fd998e"> +<li><a id="org75b2f51"></a>(23,5)3<br /> +<div class="outline-text-5" id="text-org75b2f51"> <p> 3 x 3° + 2 x 3 + 5 x 3¯¹ = (10.67)10<br /> </p> </div> <ul class="org-ul"> -<li><a id="orgadae7f4"></a>(10,67)10 -—> (2)<br /> -<div class="outline-text-6" id="text-orgadae7f4"> +<li><a id="orgbc5bb6a"></a>(10,67)10 -—> (2)<br /> +<div class="outline-text-6" id="text-orgbc5bb6a"> <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><br /> </p> @@ -779,8 +779,8 @@ En utilisant le tableau 3bits:<br /> </p> </div> </li> -<li><a id="org67ed4a9"></a>(001 010,101)2 -—> (8)<br /> -<div class="outline-text-6" id="text-org67ed4a9"> +<li><a id="org0edf1dd"></a>(001 010,101)2 -—> (8)<br /> +<div class="outline-text-6" id="text-org0edf1dd"> <p> <b>Ô Magic 3bits table, save me soul, me children and me maiden:</b><br /> </p> @@ -791,8 +791,8 @@ En utilisant le tableau 3bits:<br /> </p> </div> </li> -<li><a id="org74cd6b0"></a>(10,67)10 -—> (16)<br /> -<div class="outline-text-6" id="text-org74cd6b0"> +<li><a id="org58aafdf"></a>(10,67)10 -—> (16)<br /> +<div class="outline-text-6" id="text-org58aafdf"> <p> 10/16 = 0 R <b>A</b><br /> </p> @@ -812,13 +812,13 @@ En utilisant le tableau 3bits:<br /> </li> </ul> </div> -<div id="outline-container-org33da6ab" class="outline-4"> -<h4 id="org33da6ab">Exo2:</h4> -<div class="outline-text-4" id="text-org33da6ab"> +<div id="outline-container-org6654eb2" class="outline-4"> +<h4 id="org6654eb2">Exo2:</h4> +<div class="outline-text-4" id="text-org6654eb2"> </div> <ul class="org-ul"> -<li><a id="org98abd58"></a>(34)? = (22)10<br /> -<div class="outline-text-5" id="text-org98abd58"> +<li><a id="org15754eb"></a>(34)? = (22)10<br /> +<div class="outline-text-5" id="text-org15754eb"> <p> (34)a = (22)10 ; 4 x a° + 3 x a = 22 ; 4 + 3a = 22 ; 3a = 18<br /> </p> @@ -829,8 +829,8 @@ En utilisant le tableau 3bits:<br /> </p> </div> </li> -<li><a id="orge128996"></a>(75)? = (117)10<br /> -<div class="outline-text-5" id="text-orge128996"> +<li><a id="org1c4cbd0"></a>(75)? = (117)10<br /> +<div class="outline-text-5" id="text-org1c4cbd0"> <p> (75)b = (117)10 ; 5 x b° + 7 x b¹ = 117 ; 5 + 7b = 117 ; 7b = 112<br /> </p> @@ -843,27 +843,27 @@ En utilisant le tableau 3bits:<br /> </li> </ul> </div> -<div id="outline-container-org401e1ed" class="outline-4"> -<h4 id="org401e1ed">Exo3:</h4> -<div class="outline-text-4" id="text-org401e1ed"> +<div id="outline-container-org9da39d0" class="outline-4"> +<h4 id="org9da39d0">Exo3:</h4> +<div class="outline-text-4" id="text-org9da39d0"> </div> <ul class="org-ul"> -<li><a id="org26f3ce6"></a>(101011)2 + (111011)2<br /> -<div class="outline-text-5" id="text-org26f3ce6"> +<li><a id="org9098512"></a>(101011)2 + (111011)2<br /> +<div class="outline-text-5" id="text-org9098512"> <p> 101011 + 111011 = 1100110<br /> </p> </div> </li> -<li><a id="org3de5c4a"></a>(1011,1101)2 + (11,1)2<br /> -<div class="outline-text-5" id="text-org3de5c4a"> +<li><a id="org390626e"></a>(1011,1101)2 + (11,1)2<br /> +<div class="outline-text-5" id="text-org390626e"> <p> 1011,1101 + 11,1000 = 1111,0101<br /> </p> </div> </li> -<li><a id="orga1c43e5"></a>(1010,0101)2 - (110,1001)2<br /> -<div class="outline-text-5" id="text-orga1c43e5"> +<li><a id="org3d08c66"></a>(1010,0101)2 - (110,1001)2<br /> +<div class="outline-text-5" id="text-org3d08c66"> <p> 1010,0101 - 110,1001 = 11,1100<br /> </p> @@ -872,13 +872,13 @@ En utilisant le tableau 3bits:<br /> </ul> </div> </div> -<div id="outline-container-org68f9820" class="outline-3"> -<h3 id="org68f9820">L’arithmétique binaire (Suite): <i>Oct 4</i></h3> -<div class="outline-text-3" id="text-org68f9820"> +<div id="outline-container-org2039fb1" class="outline-3"> +<h3 id="org2039fb1">L’arithmétique binaire (Suite): <i>Oct 4</i></h3> +<div class="outline-text-3" id="text-org2039fb1"> </div> -<div id="outline-container-org69a2f51" class="outline-4"> -<h4 id="org69a2f51">La multiplication :</h4> -<div class="outline-text-4" id="text-org69a2f51"> +<div id="outline-container-orgfc45d53" class="outline-4"> +<h4 id="orgfc45d53">La multiplication :</h4> +<div class="outline-text-4" id="text-orgfc45d53"> <p> 0 x 0 = 0<br /> </p> @@ -899,9 +899,9 @@ En utilisant le tableau 3bits:<br /> </p> </div> </div> -<div id="outline-container-org62ef065" class="outline-4"> -<h4 id="org62ef065">La division :</h4> -<div class="outline-text-4" id="text-org62ef065"> +<div id="outline-container-org91fcccc" class="outline-4"> +<h4 id="org91fcccc">La division :</h4> +<div class="outline-text-4" id="text-org91fcccc"> <p> On divise de la manière la plus normale du monde !!!<br /> </p> @@ -909,49 +909,49 @@ On divise de la manière la plus normale du monde !!!<br /> </div> </div> </div> -<div id="outline-container-org12d79ca" class="outline-2"> -<h2 id="org12d79ca">4th cours : Le codage <i>Oct 10</i></h2> -<div class="outline-text-2" id="text-org12d79ca"> +<div id="outline-container-orgcbf8da9" class="outline-2"> +<h2 id="orgcbf8da9">4th cours : Le codage <i>Oct 10</i></h2> +<div class="outline-text-2" id="text-orgcbf8da9"> </div> -<div id="outline-container-orgd5b1107" class="outline-3"> -<h3 id="orgd5b1107">Le codage des entiers positifs</h3> -<div class="outline-text-3" id="text-orgd5b1107"> +<div id="outline-container-org34693c1" class="outline-3"> +<h3 id="org34693c1">Le codage des entiers positifs</h3> +<div class="outline-text-3" id="text-org34693c1"> <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)]<br /> </p> </div> </div> -<div id="outline-container-orgbc351c0" class="outline-3"> -<h3 id="orgbc351c0">Le codage des nombres relatifs</h3> -<div class="outline-text-3" id="text-orgbc351c0"> +<div id="outline-container-org3c8ed5c" class="outline-3"> +<h3 id="org3c8ed5c">Le codage des nombres relatifs</h3> +<div class="outline-text-3" id="text-org3c8ed5c"> </div> -<div id="outline-container-org2110907" class="outline-4"> -<h4 id="org2110907">Remarque</h4> -<div class="outline-text-4" id="text-org2110907"> +<div id="outline-container-orgb2d4951" class="outline-4"> +<h4 id="orgb2d4951">Remarque</h4> +<div class="outline-text-4" id="text-orgb2d4951"> <p> Quelque soit le codage utilise, par convention le dernier bit est reserve pour le signe. ou 1 est negatif et 0 est positif.<br /> </p> </div> </div> -<div id="outline-container-org3c534f2" class="outline-4"> -<h4 id="org3c534f2">Le codage en signe + valeur absolue (SVA):</h4> -<div class="outline-text-4" id="text-org3c534f2"> +<div id="outline-container-orgca1d761" class="outline-4"> +<h4 id="orgca1d761">Le codage en signe + valeur absolue (SVA):</h4> +<div class="outline-text-4" id="text-orgca1d761"> <p> Avec n bits le n eme est reserve au signe : [-(2^n-1)-1 , 2^n-1 -1]. Sur 8bits [-127, 127]<br /> </p> </div> </div> -<div id="outline-container-org0259b15" class="outline-4"> -<h4 id="org0259b15">Codage en compliment a 1 (CR):</h4> -<div class="outline-text-4" id="text-org0259b15"> +<div id="outline-container-orgd2c678f" class="outline-4"> +<h4 id="orgd2c678f">Codage en compliment a 1 (CR):</h4> +<div class="outline-text-4" id="text-orgd2c678f"> <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)<br /> </p> </div> </div> -<div id="outline-container-orgf83cdec" class="outline-4"> -<h4 id="orgf83cdec">Codage en compliment a 2 (CV):</h4> -<div class="outline-text-4" id="text-orgf83cdec"> +<div id="outline-container-org9961620" class="outline-4"> +<h4 id="org9961620">Codage en compliment a 2 (CV):</h4> +<div class="outline-text-4" id="text-org9961620"> <p> C’est literallement CR + 1 pour les negatifs et SVA pour les nombres positifs<br /> </p> @@ -962,7 +962,7 @@ C’est literallement CR + 1 pour les negatifs et SVA pour les nombres posit </div> <div id="postamble" class="status"> <p class="author">Author: Crystal</p> -<p class="date">Created: 2023-11-01 Wed 20:10</p> +<p class="date">Created: 2023-11-01 Wed 20:16</p> </div> </body> </html> \ No newline at end of file |
