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diff --git a/uni_notes/algebra.html b/uni_notes/algebra.html new file mode 100755 index 0000000..27c8d76 --- /dev/null +++ b/uni_notes/algebra.html @@ -0,0 +1,1295 @@ +<?xml version="1.0" encoding="utf-8"?> +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" +"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> +<head> +<!-- 2023-10-11 Wed 19:04 --> +<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> +<meta name="viewport" content="width=device-width, initial-scale=1" /> +<title>Algebra 1</title> +<meta name="author" content="Crystal" /> +<meta name="generator" content="Org Mode" /> +<link rel="stylesheet" type="text/css" href="../src/css/colors.css"/> +<link rel="stylesheet" type="text/css" href="../src/css/style.css"/> +<script> + window.MathJax = { + tex: { + ams: { + multlineWidth: '85%' + }, + tags: 'ams', + tagSide: 'right', + tagIndent: '.8em' + }, + chtml: { + scale: 1.0, + displayAlign: 'center', + displayIndent: '0em' + }, + svg: { + scale: 1.0, + displayAlign: 'center', + displayIndent: '0em' + }, + output: { + font: 'mathjax-modern', + displayOverflow: 'overflow' + } + }; +</script> + +<script + id="MathJax-script" + async + src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"> +</script> +</head> +<body> +<div id="content" class="content"> +<h1 class="title">Algebra 1</h1> +<div id="outline-container-orgcc5f4d8" class="outline-2"> +<h2 id="orgcc5f4d8">Contenu de la Matiére</h2> +<div class="outline-text-2" id="text-orgcc5f4d8"> +</div> +<div id="outline-container-orgf4040f2" class="outline-3"> +<h3 id="orgf4040f2">Rappels et compléments (11H)</h3> +<div class="outline-text-3" id="text-orgf4040f2"> +<ul class="org-ul"> +<li>Logique mathématique et méthodes du raisonnement mathématique</li> +<li>Ensembles et Relations</li> +<li>Applications</li> +</ul> +</div> +</div> +<div id="outline-container-orge74cdc9" class="outline-3"> +<h3 id="orge74cdc9">Structures Algébriques (11H)</h3> +<div class="outline-text-3" id="text-orge74cdc9"> +<ul class="org-ul"> +<li>Groupes et morphisme de groupes</li> +<li>Anneaux et morphisme d’anneaux</li> +<li>Les corps</li> +</ul> +</div> +</div> +<div id="outline-container-org0b54650" class="outline-3"> +<h3 id="org0b54650">Polynômes et fractions rationnelles</h3> +<div class="outline-text-3" id="text-org0b54650"> +<ul class="org-ul"> +<li>Notion du polynôme à une indéterminée á coefficients dans un anneau</li> +<li>Opérations Algébriques sur les polynômes</li> +<li>Arithmétique dans l’anneau des polynômes</li> +<li>Polynôme dérivé et formule de Taylor</li> +<li>Notion de racine d’un polynôme</li> +<li>Notion de Fraction rationelle á une indéterminée</li> +<li>Décomposition des fractions rationelles en éléments simples</li> +</ul> +</div> +</div> +</div> +<div id="outline-container-org9dbc8bb" class="outline-2"> +<h2 id="org9dbc8bb">Premier cours : Logique mathématique et méthodes du raisonnement mathématique <i>Sep 25</i> :</h2> +<div class="outline-text-2" id="text-org9dbc8bb"> +<p> +Let <b>P</b> <b>Q</b> and <b>R</b> be propositions which can either be <b>True</b> or <b>False</b>. And let’s also give the value <b>1</b> to each <b>True</b> proposition and <b>0</b> to each false one. +</p> + +<p> +<i>Ex:</i> +</p> +<ul class="org-ul"> +<li><b>5 ≥ 2</b> is a proposition, a correct one !!!</li> +<li><b>The webmaster is a girl</b> is also a proposition, which is also correct.</li> +<li><b>x is always bigger than 5</b> is <b>not</b> a proposition, because we CAN’T determine if it’s correct or not as <b>x</b> changes.</li> +</ul> +<p> +…etc +</p> + +<p> +In order to avoid repetition, and rewriting the proposition over and over, we just assign a capital letter to them such as <b>P Q</b> or <b>R</b>. +</p> + +<p> +So now we could write : +<b>Let the proposition P be 5 ≥ 2, we notice that P is always True, therefor its validity is 1</b> +</p> + +<p> +We also have the opposite of <b>P</b>, which is <b>not(P)</b> but for simplicity we use <b>P̅</b> (A P with a bar on top, in case it doesn’t load for you), now let’s go back to the previous example: +</p> + +<p> +<b>Since we know that the proposition P is true, we can conclude that P̅ is false. As P and P̅ can NOT be true at the same time. It’s like saying 5 is greater and also lesser than 2…doesn’t make sense, does it ?</b> +</p> + +<p> +Now let’s say we have two propositions, and we want to test the validity of their disjunction….. Okay what is this “disjunction” ? <b>Great Question Billy !!!</b> A disjunction is true if either propositions are true +</p> + +<p> +Ex: +<b>Let proposition P be “The webmaster is asleep”, and Q be “The reader loves pufferfishes”. The disjunction of these two propositions can have 4 different values showed in this Table of truth (such a badass name):</b> +</p> + +<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> + + +<colgroup> +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> +</colgroup> +<thead> +<tr> +<th scope="col" class="org-right">P</th> +<th scope="col" class="org-right">Q</th> +<th scope="col" class="org-right">Disjunction</th> +</tr> +</thead> +<tbody> +<tr> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +</tr> + +<tr> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +</tr> + +<tr> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +</tr> + +<tr> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +</tr> +</tbody> +</table> + +<p> +<i>What the hell is this ?</i> +The first colomn is equivalent to saying : “The webmaster is asleep AND The reader loves pufferfishes” +The second one means : “The webmaster is asleep AND The reader DOESN’T love pufferfishes (if you are in this case, then <b>I HATE YOU</b>)” +The third one… <i>zzzzzzz</i> +</p> + +<p> +You got the idea !!! +And since we are talking about a disjunction here, <b>one of the propositions</b> need to be true in order for this disjunction to be true. +</p> + +<p> +You may be wondering…. Crystal, can’t we write a disjunction in magical math symbols ? And to this I respond with a big <b>YES</b>. A disjunction is symbolized by a <b>∨</b> . So the disjunction between proposition <b>P & Q</b> can be written this way : <b>P ∨ Q</b> +</p> + +<p> +What if, we want to test whether or not two propositions are true AT THE SAME TIME ? Long story short, we can, it’s called a conjunction, same concept, as before, only this time the symbol is <b>P ∧ Q</b>, and is only true if <b>P</b> and <b>Q</b> are true. So we get a Table like this : +</p> + +<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> + + +<colgroup> +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> +</colgroup> +<thead> +<tr> +<th scope="col" class="org-right">P</th> +<th scope="col" class="org-right">Q</th> +<th scope="col" class="org-right">P ∨ Q</th> +<th scope="col" class="org-right">P ∧ Q</th> +</tr> +</thead> +<tbody> +<tr> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +</tr> + +<tr> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +</tr> + +<tr> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +</tr> + +<tr> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +</tr> +</tbody> +</table> + +<p> +<b>Always remember: 1 means true and 0 means false</b> +</p> + +<p> +There are two more basics to cover here before going to some properties, the first one is implication symbolized by the double arrow <b>⇒</b> +</p> + +<p> +Implication is kinda hard for my little brain to explain, so I will just say what it means: +</p> + +<p> +<b>If P implies Q, this means that either Q, or the opposite of P are correct</b> +</p> + +<p> +or in math terms +</p> + +<p> +<b>P ⇒ Q translates to P̅ ∨ Q</b> +Let’s illustrate : +</p> + +<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> + + +<colgroup> +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> +</colgroup> +<thead> +<tr> +<th scope="col" class="org-right">P</th> +<th scope="col" class="org-right">Q</th> +<th scope="col" class="org-right">P̅</th> +<th scope="col" class="org-right">Q̅</th> +<th scope="col" class="org-right">P ∨ Q</th> +<th scope="col" class="org-right">P ∧ Q</th> +<th scope="col" class="org-right">P ⇒ Q (P̅ ∨ Q)</th> +</tr> +</thead> +<tbody> +<tr> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +</tr> + +<tr> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +</tr> + +<tr> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +</tr> + +<tr> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +</tr> +</tbody> +</table> + +<p> +<b>If you look clearly, there is only one case where an implication is false. therefor you just need to find it, and blindly say that the others are correct. A rule of thumb is that: “A correct never implies a false”, or “If a 1 tries to imply a 0, the implication is a 0”</b> +</p> + +<p> +Aight, a last one and we are done!!! Equivalence, which is fairly easy, symbolized by a <b>⇔</b> symbol. +</p> + +<p> +A proposition is equivalent to another only when both of them have <b>the same value of truth</b> AKA: both true or both false. a little table will help demonstrate what i mean. +</p> + +<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> + + +<colgroup> +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> +</colgroup> +<thead> +<tr> +<th scope="col" class="org-right">P</th> +<th scope="col" class="org-right">Q</th> +<th scope="col" class="org-right">P̅</th> +<th scope="col" class="org-right">Q̅</th> +<th scope="col" class="org-right">P ∨ Q</th> +<th scope="col" class="org-right">P ∧ Q</th> +<th scope="col" class="org-right">P ⇒ Q (P̅ ∨ Q)</th> +<th scope="col" class="org-right">P ⇔ Q</th> +</tr> +</thead> +<tbody> +<tr> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +</tr> + +<tr> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +</tr> + +<tr> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +</tr> + +<tr> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +</tr> +</tbody> +</table> + +<p> +<i>Note: P implying Q is equivalent to P̅ implying Q̅, or: (P ⇒ Q) ⇔ (P̅ ⇒ Q̅)</i> +</p> +</div> +<div id="outline-container-org29099d5" class="outline-3"> +<h3 id="org29099d5">Properties:</h3> +<div class="outline-text-3" id="text-org29099d5"> +</div> +<div id="outline-container-orgfeb62c8" class="outline-4"> +<h4 id="orgfeb62c8"><b>Absorption</b>:</h4> +<div class="outline-text-4" id="text-orgfeb62c8"> +<p> +(P ∨ P) ⇔ P +</p> + +<p> +(P ∧ P) ⇔ P +</p> +</div> +</div> +<div id="outline-container-org2fa2b1f" class="outline-4"> +<h4 id="org2fa2b1f"><b>Commutativity</b>:</h4> +<div class="outline-text-4" id="text-org2fa2b1f"> +<p> +(P ∧ Q) ⇔ (Q ∧ P) +</p> + +<p> +(P ∨ Q) ⇔ (Q ∨ P) +</p> +</div> +</div> +<div id="outline-container-orge30be26" class="outline-4"> +<h4 id="orge30be26"><b>Associativity</b>:</h4> +<div class="outline-text-4" id="text-orge30be26"> +<p> +P ∧ (Q ∧ R) ⇔ (P ∧ Q) ∧ R +</p> + +<p> +P ∨ (Q ∨ R) ⇔ (P ∨ Q) ∨ R +</p> +</div> +</div> +<div id="outline-container-orgd1de8d5" class="outline-4"> +<h4 id="orgd1de8d5"><b>Distributivity</b>:</h4> +<div class="outline-text-4" id="text-orgd1de8d5"> +<p> +P ∧ (Q ∨ R) ⇔ (P ∧ Q) ∨ (P ∧ R) +</p> + +<p> +P ∨ (Q ∧ R) ⇔ (P ∨ Q) ∧ (P ∨ R) +</p> +</div> +</div> +<div id="outline-container-org8c5c9f6" class="outline-4"> +<h4 id="org8c5c9f6"><b>Neutral element</b>:</h4> +<div class="outline-text-4" id="text-org8c5c9f6"> +<p> +<i>We define proposition <b>T</b> to be always <b>true</b> and <b>F</b> to be always <b>false</b></i> +</p> + +<p> +P ∧ T ⇔ P +</p> + +<p> +P ∨ F ⇔ P +</p> +</div> +</div> +<div id="outline-container-orgf102dde" class="outline-4"> +<h4 id="orgf102dde"><b>Negation of a conjunction & a disjunction</b>:</h4> +<div class="outline-text-4" id="text-orgf102dde"> +<p> +Now we won’t use bars here because my lazy ass doesn’t know how, so instead I will use not()!!! +</p> + +<p> +not(<b>P ∧ Q</b>) ⇔ P̅ ∨ Q̅ +</p> + +<p> +not(<b>P ∨ Q</b>) ⇔ P̅ ∧ Q̅ +</p> + +<p> +<b>A rule I really like to use here is: Break and Invert. Basically you break the bar into the three characters of the propositions, so you get not(P) not(∧ or ∨) <i>NOT AN ACTUAL MATH WRITING. DONT USE IT ANYWHERE ELSE OTHER THAN YOUR BRAIN</i> and not(Q)</b> +</p> +</div> +</div> +<div id="outline-container-org9534ece" class="outline-4"> +<h4 id="org9534ece"><b>Transitivity</b>:</h4> +<div class="outline-text-4" id="text-org9534ece"> +<p> +[(P ⇒ Q) (Q ⇒ R)] ⇔ P ⇒ R +</p> +</div> +</div> +<div id="outline-container-org90fa987" class="outline-4"> +<h4 id="org90fa987"><b>Contraposition</b>:</h4> +<div class="outline-text-4" id="text-org90fa987"> +<p> +(P ⇒ Q) ⇔ (Q̅ ⇒ P̅) +</p> +</div> +</div> +<div id="outline-container-orga2d0ece" class="outline-4"> +<h4 id="orga2d0ece">God only knows what this property is called:</h4> +<div class="outline-text-4" id="text-orga2d0ece"> +<p> +<i>If</i> +</p> + +<p> +(P ⇒ Q) is true +</p> + +<p> +and +</p> + +<p> +(Q̅ ⇒ Q) is true +</p> + +<p> +then +</p> + +<p> +Q is always true +</p> +</div> +</div> +</div> +<div id="outline-container-org35a43b7" class="outline-3"> +<h3 id="org35a43b7">Some exercices I found online :</h3> +<div class="outline-text-3" id="text-org35a43b7"> +</div> +<div id="outline-container-orgf619324" class="outline-4"> +<h4 id="orgf619324">USTHB 2022/2023 Section B :</h4> +<div class="outline-text-4" id="text-orgf619324"> +</div> +<ul class="org-ul"> +<li><a id="org1c47389"></a>Exercice 1: Démontrer les équivalences suivantes:<br /> +<div class="outline-text-5" id="text-org1c47389"> +<ol class="org-ol"> +<li><p> +(P ⇒ Q) ⇔ (Q̅ ⇒ P̅) +</p> + +<p> +Basically we are asked to prove contraposition, so here we have ( P ⇒ Q ) which is equivalent to P̅ ∨ Q <b>By definition : (P ⇒ Q) ⇔ (P̅ ∨ Q)</b> +</p></li> +</ol> + + +<p> +So we end up with : <b>(P̅ ∨ Q) ⇔ (Q̅ ⇒ P̅)</b>, now we just do the same with the second part of the contraposition. <b>(Q̅ ⇒ P̅) ⇔ (Q ∨ P̅)</b> therefor : +</p> + + +<p> +<b>(Q ∨ P̅) ⇔ (P̅ ∨ Q)</b>, which is true because of commutativity +</p> + +<ol class="org-ol"> +<li>not(P ⇒ Q) ⇔ P ∧ Q̅</li> +</ol> + + +<p> +Okaaaay so, let’s first get rid of the implication, because I don’t like it : <b>not(P̅ ∨ Q)</b> +</p> + + +<p> +Now that we got rid of it, we can negate the whole disjunction <b>not(P̅ ∨ Q) ⇔ (P ∧ Q̅)</b>. Which is the equivalence we needed to prove +</p> + +<ol class="org-ol"> +<li><p> +P ⇒ (Q ∧ R) ⇔ (P ⇒ Q) ∧ (P ⇒ R) +</p> + +<p> +One might be tempted to replace P with P̅ to get rid of the implication…sadly this isnt it. All we have to do here is resort to <b>Distributivity</b>, because yeah, we can distribute an implication across a {con/dis}junction +</p></li> + +<li><p> +P ∧ (Q ∨ R) ⇔ (P ∧ Q) ∨ (P ∧ R) +</p> + +<p> +Literally the same as above 🩷 +</p></li> +</ol> +</div> +</li> +<li><a id="org688fdcc"></a>Exercice 2: Dire si les propositions suivantes sont vraies ou fausses, et les nier:<br /> +<div class="outline-text-5" id="text-org688fdcc"> +<ol class="org-ol"> +<li><p> +∀x ∈ ℝ ,∃y ∈ ℝ*+, tels que e^x = y +</p> + +<p> +For each x from the set of Real numbers, there exists a number y from the set of non-zero positive Real numbers that satisfies the equation : e^x = y +</p></li> +</ol> + + +<p> +“The function f(x)=e^x is always positive and non-null”, the very definition of an exponential function !!!! +</p> + + +<p> +<b>So the proposition is true</b> +</p> + + +<ol class="org-ol"> +<li>∃x ∈ ℝ, tels que x^2 < x < x^3</li> +</ol> + + +<p> +We just need to find a value that satisifies this condition…thankfully its easy…. +</p> + +<p> +x² < x < x³ , we divide the three terms by x so we get : +</p> + + +<p> +x < 1 < x² , or : +</p> + + +<p> +<b>x < 1</b> ; <b>1 < x²</b> ⇔ <b>x < 1</b> ; <b>1 < x</b> <i>We square root both sides</i> +</p> + + +<p> +We end up with a contradiction, therefor its wrong +</p> + + +<ol class="org-ol"> +<li>∀x ∈ ℝ, ∃y ∈ ℝ tels que y = 3x - 8</li> +</ol> + + +<p> +I dont really understand this one, so let me translate it “For any value of x from the set of Real numbers, 3x - 8 is a Real number”…. i mean….yeah, we are substracting a Real number from an other real number… +</p> + +<p> +<b>Since substraction is an Internal composition law in ℝ, therefor all results of a substraction between two Real numbers is…Real</b> +</p> + +<ol class="org-ol"> +<li><p> +∃x ∈ ℕ, ∀y ∈ ℕ, x > y ⇒ x + y < 8 +</p> + +<p> +“There exists a number x from the set of Natural numbers such as for all values of y from the set of Natural numbers, x > y implies x + y < 8” +</p></li> +</ol> + + +<p> +Let’s get rid of the implication : +</p> + +<p> +∃x ∈ ℕ, ∀y ∈ ℕ, (y > x) ∨ (x + y < 8) <i>There exists a number x from the set of Natural numbers such as for all values of y from the set of Natural numbers y > x OR x + y < 8</i> +</p> + +<p> +This proposition is true, because there exists a value of x that satisfies this condition, it’s <b>all numbers under 8</b> let’s take 3 as an example: +</p> + + +<p> +<b>x = 3 , if y > 3 then the first condition is true ; if y < 3 then the second one is true</b> +</p> + + +<p> +Meaning that the two propositions CAN NOT BE WRONG TOGETHER, either one is wrong, or the other +</p> + + +<p> +y > x +</p> + + +<p> +<b>y - x > 0</b> +</p> + + +<p> +y + x < 8 +</p> + + +<p> +<b>y < 8 - x</b> <i>This one is always true for all values of x below 8, since we are working in the set ℕ</i> +</p> + + +<ol class="org-ol"> +<li><p> +∀x ∈ ℝ, x² ≥ 1 ⇔ x ≥ 1 +</p> + +<p> +….This is getting stupid. of course it’s true it’s part of the definition of the power of 2 +</p></li> +</ol> +</div> +</li> +</ul> +</div> +</div> +</div> +<div id="outline-container-orgac834f2" class="outline-2"> +<h2 id="orgac834f2">2éme cours <i>Oct 2</i></h2> +<div class="outline-text-2" id="text-orgac834f2"> +</div> +<div id="outline-container-org42a8fad" class="outline-3"> +<h3 id="org42a8fad">Quantifiers</h3> +<div class="outline-text-3" id="text-org42a8fad"> +<p> +A propriety P can depend on a parameter x +</p> + + +<p> +∀ is the universal quantifier which stands for “For any value of…” +</p> + + +<p> +∃ is the existential quantifier which stands for “There exists at least one…” +</p> +</div> +<ul class="org-ul"> +<li><a id="org48c4773"></a>Example<br /> +<div class="outline-text-6" id="text-org48c4773"> +<p> +P(x) : x+1≥0 +</p> + +<p> +P(X) is True or False depending on the values of x +</p> +</div> +</li> +</ul> +<div id="outline-container-org923b5fe" class="outline-4"> +<h4 id="org923b5fe">Proprieties</h4> +<div class="outline-text-4" id="text-org923b5fe"> +</div> +<ul class="org-ul"> +<li><a id="orgaf93dd2"></a>Propriety Number 1:<br /> +<div class="outline-text-5" id="text-orgaf93dd2"> +<p> +The negation of the universal quantifier is the existential quantifier, and vice-versa : +</p> + +<ul class="org-ul"> +<li>not(∀x ∈ E , P(x)) ⇔ ∃ x ∈ E, not(P(x))</li> +<li>not(∃x ∈ E , P(x)) ⇔ ∀ x ∈ E, not(P(x))</li> +</ul> +</div> +<ul class="org-ul"> +<li><a id="orgf4f038f"></a>Example:<br /> +<div class="outline-text-6" id="text-orgf4f038f"> +<p> +∀ x ≥ 1 x² > 5 ⇔ ∃ x ≥ 1 x² < 5 +</p> +</div> +</li> +</ul> +</li> +<li><a id="org2c00dae"></a>Propriety Number 2:<br /> +<div class="outline-text-5" id="text-org2c00dae"> +<p> +<b>∀x ∈ E, [P(x) ∧ Q(x)] ⇔ [∀ x ∈ E, P(x)] ∧ [∀ x ∈ E, Q(x)]</b> +</p> + + +<p> +The propriety “For any value of x from a set E , P(x) and Q(x)” is equivalent to “For any value of x from a set E, P(x) AND for any value of x from a set E, Q(x)” +</p> +</div> +<ul class="org-ul"> +<li><a id="orga95ad05"></a>Example :<br /> +<div class="outline-text-6" id="text-orga95ad05"> +<p> +P(x) : sqrt(x) > 0 ; Q(x) : x ≥ 1 +</p> + + +<p> +∀x ∈ ℝ*+, [sqrt(x) > 0 , x ≥ 1] ⇔ [∀x ∈ R*+, sqrt(x) > 0] ∧ [∀x ∈ R*+, x ≥ 1] +</p> + + +<p> +<b>Which is true</b> +</p> +</div> +</li> +</ul> +</li> +<li><a id="orgfe5ddf2"></a>Propriety Number 3:<br /> +<div class="outline-text-5" id="text-orgfe5ddf2"> +<p> +<b>∃ x ∈ E, [P(x) ∧ Q(x)] <i>⇒</i> [∃ x ∈ E, P(x)] ∧ [∃ x ∈ E, Q(x)]</b> +</p> + + +<p> +<i>Here its an implication and not an equivalence</i> +</p> +</div> +<ul class="org-ul"> +<li><a id="orgca7bea7"></a>Example of why it’s NOT an equivalence :<br /> +<div class="outline-text-6" id="text-orgca7bea7"> +<p> +P(x) : x > 5 ; Q(x) : x < 5 +</p> + + +<p> +Of course there is no value of x such as its inferior and superior to 5 at the same time, so obviously the proposition is false. However, the two propositions separated are correct on their own, because there is a value of x such as its superior to 5, and there is also a value of x such as its inferior to 5. This is why it’s an implication and NOT AN EQUIVALENCE!!! +</p> +</div> +</li> +</ul> +</li> +<li><a id="orgf4ecdc0"></a>Propriety Number 4:<br /> +<div class="outline-text-5" id="text-orgf4ecdc0"> +<p> +<b>[∀ x ∈ E, P(x)] ∨ [∀ x ∈ E, Q(x)] <i>⇒</i> ∀x ∈ E, [P(x) ∨ Q(x)]</b> +</p> + + +<p> +<i>Same here, implication and NOT en equivalence</i> +</p> +</div> +</li> +</ul> +</div> +</div> +<div id="outline-container-org6421557" class="outline-3"> +<h3 id="org6421557">Multi-parameter proprieties :</h3> +<div class="outline-text-3" id="text-org6421557"> +<p> +A propriety P can depend on two or more parameters, for convenience we call them x,y,z…etc +</p> +</div> +<ul class="org-ul"> +<li><a id="org314cff3"></a>Example :<br /> +<div class="outline-text-6" id="text-org314cff3"> +<p> +P(x,y): x+y > 0 +</p> + + +<p> +P(0,1) is a True proposition +</p> + + +<p> +P(-2,-1) is a False one +</p> +</div> +</li> +<li><a id="orga9b6089"></a>WARNING :<br /> +<div class="outline-text-6" id="text-orga9b6089"> +<p> +∀x ∈ E, ∃y ∈ F , P(x,y) +</p> + + +<p> +∃y ∈ F, ∀x ∈ E , P(x,y) +</p> + + +<p> +Are different because in the first one y depends on x, while in the second one, it doesn’t +</p> +</div> +<ul class="org-ul"> +<li><a id="org3c5e5a8"></a>Example :<br /> +<div class="outline-text-7" id="text-org3c5e5a8"> +<p> +∀ x ∈ ℕ , ∃ y ∈ ℕ y > x -–— True +</p> + + +<p> +∃ y ∈ ℕ , ∀ x ∈ ℕ y > x -–— False +</p> +</div> +</li> +</ul> +</li> +</ul> +<li><a id="org2f3208e"></a>Proprieties :<br /> +<div class="outline-text-5" id="text-org2f3208e"> +<ol class="org-ol"> +<li>not(∀x ∈ E ,∃y ∈ F P(x,y)) ⇔ ∃x ∈ E, ∀y ∈ F not(P(x,y))</li> +<li>not(∃x ∈ E ,∀y ∈ F P(x,y)) ⇔ ∀x ∈ E, ∃y ∈ F not(P(x,y))</li> +</ol> +</div> +</li> +</ul> +</div> +<div id="outline-container-orgc25dd7a" class="outline-3"> +<h3 id="orgc25dd7a">Methods of mathematical reasoning :</h3> +<div class="outline-text-3" id="text-orgc25dd7a"> +</div> +<div id="outline-container-orgf843851" class="outline-4"> +<h4 id="orgf843851">Direct reasoning :</h4> +<div class="outline-text-4" id="text-orgf843851"> +<p> +To show that an implication P ⇒ Q is true, we suppose that P is true and we show that Q is true +</p> +</div> +<ul class="org-ul"> +<li><a id="org6dd2136"></a>Example:<br /> +<div class="outline-text-5" id="text-org6dd2136"> +<p> +Let a,b be two Real numbers, we have to prove that <b>a² + b² = 1 ⇒ |a + b| ≤ 2</b> +</p> + + +<p> +We suppose that a²+b² = 1 and we prove that |a + b| ≤ 2 +</p> + + +<p> +a²+b²=1 ⇒ b² = 1 - a² ; a² = 1 - b² +</p> + + +<p> +a²+b²=1 ⇒ 1 - a² ≥ 0 ; 1 - b² ≥ 0 +</p> + + +<p> +a²+b²=1 ⇒ a² ≤ 1 ; b² ≤ 1 +</p> + + +<p> +a²+b²=1 ⇒ -1 ≤ a ≤ 1 ; -1 ≤ b ≤ 1 +</p> + + +<p> +a²+b²=1 ⇒ -2 ≤ a + b ≤ 2 +</p> + + +<p> +a²+b²=1 ⇒ |a + b| ≤ 2 <b>Which is what we wanted to prove, therefor the implication is correct</b> +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-orga59c0ad" class="outline-4"> +<h4 id="orga59c0ad">Reasoning by the Absurd:</h4> +<div class="outline-text-4" id="text-orga59c0ad"> +<p> +To prove that a proposition is True, we suppose that it’s False and we must come to a contradiction +</p> + + +<p> +And to prove that an implication P ⇒ Q is true using the reasoning by the absurd, we suppose that P ∧ not(Q) is true, and then we come to a contradiction as well +</p> +</div> +<ul class="org-ul"> +<li><a id="orgf8b9f83"></a>Example:<br /> +<div class="outline-text-5" id="text-orgf8b9f83"> +<p> +Prove that this proposition is correct using the reasoning by the absurd : ∀x ∈ ℝ* , sqrt(1+x²) ≠ 1 + x²/2 +</p> + + +<p> +We assume that ∃ x ℝ* , sqrt(1+x²) = 1 + x²/2 +</p> + + +<p> +sqrt(1+x²) = 1 + x²/2 ; 1 + x² = (1+x²/2)² ; 1 + x² = 1 + x^4/4 + x² ; x^(4)/4 = 0 … Which contradicts with our proposition, since x = 4 and we are working on the ℝ* set +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-orgcc285c2" class="outline-4"> +<h4 id="orgcc285c2">Reasoning by contraposition:</h4> +<div class="outline-text-4" id="text-orgcc285c2"> +<p> +If an implication P ⇒ Q is too hard to prove, we just have to prove not(Q) ⇒ not(P) is true !!! or in other words that both not(P) and not(Q) are true +</p> +</div> +</div> +<div id="outline-container-org2e67808" class="outline-4"> +<h4 id="org2e67808">Reasoning by counter example:</h4> +<div class="outline-text-4" id="text-org2e67808"> +<p> +To prove that a proposition ∀x ∈ E, P(x) is false, all we have to do is find a single value of x from E such as not(P(x)) is true +</p> +</div> +</div> +</div> +</div> +<div id="outline-container-org7440601" class="outline-2"> +<h2 id="org7440601">3eme Cours : <i>Oct 9</i></h2> +<div class="outline-text-2" id="text-org7440601"> +</div> +<div id="outline-container-org70aa2db" class="outline-4"> +<h4 id="org70aa2db">Reasoning by recurrence :</h4> +<div class="outline-text-4" id="text-org70aa2db"> +<p> +P is a propriety dependent of <b>n ∈ ℕ</b>. If for n0 ∈ ℕ P(n0) is true, and if for n ≥ n0 (P(n) ⇒ P(n+1)) is true. Then P(n) is true for n ≥ n0 +</p> +</div> +<ul class="org-ul"> +<li><a id="org52e5298"></a>Example:<br /> +<div class="outline-text-5" id="text-org52e5298"> +<p> +Let’s prove that ∀ n ≥ 1 , (n,k=1)Σk = [n(n+1)]/2 +</p> + + +<p> +P(n) : (n,k=1)Σk = [n(n+1)]/2 +</p> + + + +<p> +<b>Pour n = 1:</b> (1,k=1)Σk = 1 ; [n(n+1)]/2 = 1 . <b>So P(1) is true</b> +</p> + + + +<p> +For n ≥ 1. We assume that P(n) is true, OR : <b>(n, k=1)Σk = n(n+1)/2</b>. We now have to prove that P(n+1) is true, Or : <b>(n+1, k=1)Σk = (n+1)(n+2)/2</b> +</p> + + +<p> +(n+1, k=1)Σk = 1 + 2 + …. + n + (n+1) ; (n+1, k=1)Σk = (n, k=1)Σk + (n+1) ; = n(n+1)/2 + (n+1) ; = [n(n+1) + 2(n+1)]/2 ; = <b>[(n+2)(n+1)]/2</b> <i>WHICH IS WHAT WE NEEDED TO FIND</i> +</p> + + +<p> +<b>Conclusion: ∀n ≥ 1 , (n,k=1)Σk = n(n+1)/2</b> +</p> +</div> +</li> +</ul> +</div> +</div> +<div id="outline-container-orga6a518d" class="outline-2"> +<h2 id="orga6a518d">4eme Cours : Chapitre 2 : Sets and Operations</h2> +<div class="outline-text-2" id="text-orga6a518d"> +</div> +<div id="outline-container-org31e3615" class="outline-3"> +<h3 id="org31e3615">Definition of a set :</h3> +<div class="outline-text-3" id="text-org31e3615"> +<p> +A set is a collection of objects that share the sane propriety +</p> +</div> +</div> +<div id="outline-container-orgfa9bfd1" class="outline-3"> +<h3 id="orgfa9bfd1">Belonging, inclusion, and equality :</h3> +<div class="outline-text-3" id="text-orgfa9bfd1"> +<ol class="org-ol"> +<li>Let E be a set. If x is an element of E, we say that x belongs to E we write <b>x ∈ E</b>, and if it doesn’t, we write <b>x ∉ E</b></li> +<li>A set E is included in a set F if all elements of E are elements of F and we write <b>E ⊂ F ⇔ (∀x , x ∈ E ⇒ x ∈ F)</b>. We say that E is a subset of F, or a part of F. The negation of this propriety is : <b>E ⊄ F ⇔ ∃x , x ∈ E and x ⊄ F</b></li> +<li>E and F are equal if E is included in F and F is included in E, and we write <b>E = F ⇔ (E ⊂ F) et (F ⊂ E)</b></li> +<li>The empty set (symbolized by ∅) is a set without elements, and is included in all sets (by convention) : <b>∅ ⊂ E</b></li> +</ol> +</div> +</div> +<div id="outline-container-org2a19707" class="outline-3"> +<h3 id="org2a19707">Intersections and reunions :</h3> +<div class="outline-text-3" id="text-org2a19707"> +</div> +<div id="outline-container-org6a5f566" class="outline-4"> +<h4 id="org6a5f566">Intersection:</h4> +<div class="outline-text-4" id="text-org6a5f566"> +<p> +E ∩ F = {x / x ∈ E AND x ∈ F} ; x ∈ E ∩ F ⇔ x ∈ F AND x ∈ F +</p> + + +<p> +x ∉ E ∩ F ⇔ x ∉ E OR x ∉ F +</p> +</div> +</div> +<div id="outline-container-org9bc9aeb" class="outline-4"> +<h4 id="org9bc9aeb">Union:</h4> +<div class="outline-text-4" id="text-org9bc9aeb"> +<p> +E ∪ F = {x / x ∈ E OR x ∈ F} ; x ∈ E ∪ F ⇔ x ∈ F OR x ∈ F +</p> + + +<p> +x ∉ E ∪ F ⇔ x ∉ E AND x ∉ F +</p> +</div> +</div> +<div id="outline-container-org9a7f719" class="outline-4"> +<h4 id="org9a7f719">Difference between two sets:</h4> +<div class="outline-text-4" id="text-org9a7f719"> +<p> +E\F(Which is also written as : E - F) = {x / x ∈ E and x ∉ F} +</p> +</div> +</div> +<div id="outline-container-org5f5c721" class="outline-4"> +<h4 id="org5f5c721">Complimentary set:</h4> +<div class="outline-text-4" id="text-org5f5c721"> +<p> +If F ⊂ E. E - F is the complimentary of F in E. +</p> + + +<p> +FCE = {x /x ∈ E AND x ∉ F} <b>ONLY WHEN F IS A SUBSET OF E</b> +</p> +</div> +</div> +<div id="outline-container-orga285d1d" class="outline-4"> +<h4 id="orga285d1d">Symentrical difference</h4> +<div class="outline-text-4" id="text-orga285d1d"> +<p> +E Δ F = (E - F) ∪ (F - E) ; = (E ∪ F) - (E ∩ F) +</p> +</div> +</div> +</div> +<div id="outline-container-orgc12e73b" class="outline-3"> +<h3 id="orgc12e73b">Proprieties :</h3> +<div class="outline-text-3" id="text-orgc12e73b"> +<p> +Let E,F and G be 3 sets. We have : +</p> +</div> +<div id="outline-container-org31d8697" class="outline-4"> +<h4 id="org31d8697">Commutativity:</h4> +<div class="outline-text-4" id="text-org31d8697"> +<p> +E ∩ F = F ∩ E +E ∪ F = F ∪ E +</p> +</div> +</div> +<div id="outline-container-org7080d99" class="outline-4"> +<h4 id="org7080d99">Associativity:</h4> +<div class="outline-text-4" id="text-org7080d99"> +<p> +E ∩ (F ∩ G) = (E ∩ F) ∩ G +E ∪ (F ∪ G) = (E ∪ F) ∪ G +</p> +</div> +</div> +<div id="outline-container-org13da04d" class="outline-4"> +<h4 id="org13da04d">Distributivity:</h4> +<div class="outline-text-4" id="text-org13da04d"> +<p> +E ∩ (F ∪ G) = (E ∩ F) ∪ (E ∩ G) +E ∪ (F ∩ G) = (E ∪ F) ∩ (E ∪ G) +</p> +</div> +</div> +<div id="outline-container-orgaa33b71" class="outline-4"> +<h4 id="orgaa33b71">Lois de Morgan:</h4> +<div class="outline-text-4" id="text-orgaa33b71"> +<p> +If E ⊂ G and F ⊂ G ; +</p> + +<p> +(E ∩ F)CG = ECG ∪ FCG ; (E ∪ F)CG = ECG ∩ FCG +</p> +</div> +</div> +<div id="outline-container-org7e7db42" class="outline-4"> +<h4 id="org7e7db42">An other one:</h4> +<div class="outline-text-4" id="text-org7e7db42"> +<p> +E - (F ∩ G) = (E-F) ∪ (E-G) ; E - (F ∪ G) = (E-F) ∩ (E-G) +</p> +</div> +</div> +<div id="outline-container-orgd02bd7f" class="outline-4"> +<h4 id="orgd02bd7f">An other one:</h4> +<div class="outline-text-4" id="text-orgd02bd7f"> +<p> +E ∩ ∅ = ∅ ; E ∪ ∅ = E +</p> +</div> +</div> +<div id="outline-container-org99eb39a" class="outline-4"> +<h4 id="org99eb39a">And an other one:</h4> +<div class="outline-text-4" id="text-org99eb39a"> +<p> +E ∩ (F Δ G) = (E ∩ F) Δ (E ∩ G) +</p> +</div> +</div> +<div id="outline-container-org3e9b2ef" class="outline-4"> +<h4 id="org3e9b2ef">And the last one:</h4> +<div class="outline-text-4" id="text-org3e9b2ef"> +<p> +E Δ ∅ = E ; E Δ E = ∅ +</p> +</div> +</div> +</div> +</div> +</div> +<div id="postamble" class="status"> +<p class="author">Author: Crystal</p> +<p class="date">Created: 2023-10-11 Wed 19:04</p> +</div> +</body> +</html> \ No newline at end of file diff --git a/uni_notes/alsd.html b/uni_notes/alsd.html new file mode 100755 index 0000000..9b0fa1c --- /dev/null +++ b/uni_notes/alsd.html @@ -0,0 +1,84 @@ +<?xml version="1.0" encoding="utf-8"?> +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" +"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> +<head> +<!-- 2023-10-01 Sun 20:23 --> +<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> +<meta name="viewport" content="width=device-width, initial-scale=1" /> +<title>ALSD1</title> +<meta name="author" content="Crystal" /> +<meta name="generator" content="Org Mode" /> +<link rel="stylesheet" type="text/css" href="../src/css/colors.css"/> +<link rel="stylesheet" type="text/css" href="../src/css/style.css"/> +</head> +<body> +<div id="content" class="content"> +<h1 class="title">ALSD1</h1> + +<div id="outline-container-org2b6a11d" class="outline-2"> +<h2 id="org2b6a11d">Contenu de la Matiére</h2> +<div class="outline-text-2" id="text-org2b6a11d"> +</div> +<div id="outline-container-org5340dec" class="outline-3"> +<h3 id="org5340dec">Chapitre 1: Elements de Base</h3> +<div class="outline-text-3" id="text-org5340dec"> +<ul class="org-ul"> +<li>Algorithmique, procésseur, action.</li> +<li>Programme et languages de programmation.</li> +<li>Analyse des problémes.</li> +</ul> +</div> +</div> + +<div id="outline-container-org92c03d8" class="outline-3"> +<h3 id="org92c03d8">Chapitre 2: Présentation du formalisme Algorithmique</h3> +</div> +<div id="outline-container-orgde3a680" class="outline-3"> +<h3 id="orgde3a680">Chapitre 3: Eléments de base du language C</h3> +</div> +<div id="outline-container-org4eb24e1" class="outline-3"> +<h3 id="org4eb24e1">Chapitre 4: Modularité( Fonction et Procédure )</h3> +</div> +<div id="outline-container-org6a31b85" class="outline-3"> +<h3 id="org6a31b85">Chapitre 5: Les structures des données statiques</h3> +</div> +</div> + + +<div id="outline-container-org7ccc87b" class="outline-2"> +<h2 id="org7ccc87b">Premier cours : Algorithmes <i>Oct 1</i> :</h2> +<div class="outline-text-2" id="text-org7ccc87b"> +</div> +<div id="outline-container-orge430c1a" class="outline-3"> +<h3 id="orge430c1a">Définition d’un algorithm :</h3> +<div class="outline-text-3" id="text-orge430c1a"> +<p> +Un ensemble d’opérations ecrites dans le language naturel. +</p> +</div> + +<div id="outline-container-orgbaecdfc" class="outline-4"> +<h4 id="orgbaecdfc">Example d’un Algo : Résolution d’une équation du second ordre (ax²+bx+c=0)</h4> +<div class="outline-text-4" id="text-orgbaecdfc"> +<ol class="org-ol"> +<li>Si a=0 ET b=0 alors <b>l’équation n’est pas du 2nd ordre</b>.</li> +<li>Si a=0 et b≠0 alors <b>x= -c/5</b> .</li> +<li>Si a≠0 alors <b>calculer Δ= b²-4ac</b> : +<ol class="org-ol"> +<li>Si Δ=0 alors <b>x=-b/2a</b>.</li> +<li>Si Δ<0 alors <b>l’équation n’as pas de solution</b>.</li> +<li>Si Δ>0 alors <b>x=[-b±sqrt(Δ)]/2a</b></li> +</ol></li> +</ol> +</div> +</div> +</div> +</div> +</div> +<div id="postamble" class="status"> +<p class="author">Author: Crystal</p> +<p class="date">Created: 2023-10-01 Sun 20:23</p> +</div> +</body> +</html> \ No newline at end of file diff --git a/uni_notes/analyse.html b/uni_notes/analyse.html new file mode 100755 index 0000000..20d8972 --- /dev/null +++ b/uni_notes/analyse.html @@ -0,0 +1,1123 @@ +<?xml version="1.0" encoding="utf-8"?> +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" +"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> +<head> +<!-- 2023-10-11 Wed 19:18 --> +<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> +<meta name="viewport" content="width=device-width, initial-scale=1" /> +<title>Analyse 1</title> +<meta name="author" content="Crystal" /> +<meta name="generator" content="Org Mode" /> +<link rel="stylesheet" type="text/css" href="../src/css/colors.css"/> +<link rel="stylesheet" type="text/css" href="../src/css/style.css"/> +</head> +<body> +<div id="content" class="content"> +<h1 class="title">Analyse 1</h1> +<div id="outline-container-org32ad572" class="outline-2"> +<h2 id="org32ad572">Contenu de la Matiére</h2> +<div class="outline-text-2" id="text-org32ad572"> +</div> +<div id="outline-container-org156647d" class="outline-3"> +<h3 id="org156647d">Chapitre 1 : Quelque propriétés de ℝ</h3> +<div class="outline-text-3" id="text-org156647d"> +<ul class="org-ul"> +<li>Structure algébrique de ℝ</li> +<li>L’ordre dans ℝ</li> +<li>Majorant, minorant, borne superieure, borne inférieure</li> +</ul> +</div> +</div> +<div id="outline-container-org2064013" class="outline-3"> +<h3 id="org2064013">Chapitre 2 : Les suites numériques réelles</h3> +<div class="outline-text-3" id="text-org2064013"> +<ul class="org-ul"> +<li>Définition : convergence, opérations sur les suites convergentes</li> +<li>Theoréme de convergence, Theoréme de <span class="underline">_</span> suites, sans suites, extension au limites infinies</li> +<li>Suites de cauchy, suites adjacentes et suites récurentes</li> +</ul> +</div> +</div> +<div id="outline-container-org74215b0" class="outline-3"> +<h3 id="org74215b0">Chapitre 3 : Limites et continuité des fonctions réelles d’une variable réelle</h3> +<div class="outline-text-3" id="text-org74215b0"> +<ul class="org-ul"> +<li>Les limites : définition, opérations sur les limites, les formes inditerminées</li> +<li>La continuité : définition, Theorémes fondamentaux</li> +<li>La continuité informe les fonctions Lepchitziennes</li> +</ul> +</div> +</div> +<div id="outline-container-orgfad0512" class="outline-3"> +<h3 id="orgfad0512">Chapitre 4 : La dérivabilité et son interprétation géometrique</h3> +<div class="outline-text-3" id="text-orgfad0512"> +<ul class="org-ul"> +<li>Opérations sur les fonctions dérivales, Theoréme de Rolle, Theoréme des accroissements finis, régle de L’Hopital et formule de Taylor</li> +</ul> +</div> +</div> +<div id="outline-container-org2c42f3a" class="outline-3"> +<h3 id="org2c42f3a">Chapitre 5 : Les fonctions trigonométriques réciproques, fonctions hypérboliques réciproques</h3> +<div class="outline-text-3" id="text-org2c42f3a"> +<ul class="org-ul"> +<li>Comparaison asymptotique</li> +<li>Symbole de lamdau (lambda ?), et notions des fonctions équivalentes</li> +<li>Développements limites polynominaux (D.L) et opérations sur les D.L</li> +<li>Généralisations des D.L</li> +<li>Application au calcul de limite et l’étude des branches infinies</li> +</ul> +</div> +</div> +</div> +<div id="outline-container-org4aeab0e" class="outline-2"> +<h2 id="org4aeab0e">Premier cours : Quelque propriétés de ℝ <i>Sep 26</i> :</h2> +<div class="outline-text-2" id="text-org4aeab0e"> +</div> +<div id="outline-container-org1c15646" class="outline-3"> +<h3 id="org1c15646">La loi de composition interne dans E :</h3> +<div class="outline-text-3" id="text-org1c15646"> +<p> +@ : E x E —> E + (x,y) —> x @ y +</p> + +<p> +@ est une lois de composition interne seulement si : +</p> + +<p> +<b>∀ x,y ε E</b> +</p> +</div> +<div id="outline-container-orge4f4285" class="outline-4"> +<h4 id="orge4f4285"><b>Example : Addition</b></h4> +<div class="outline-text-4" id="text-orge4f4285"> +<p> +Est ce que l’addition (+) est L.C.I dans ℕ ? +</p> + +<p> +ℕ x ℕ —> ℕ +</p> + +<p> +(x,y) —> x + y ? <i>En gros : Pour que l’addition soit une L.C.I dans ℕ, il faut que: quand on additionne <b>n’importe quel</b> chiffre x et y de N, il faut que le résultat appertiens aussi a ℕ</i> +</p> + +<p> +∀ x,y ∈ ℕ , x + y ∈ ℕ <i>En gros: Pour TOUTE valeur de x et y appartenant a ℕ, leur somme est toujours dans ℕ</i> +</p> + +<p> +Donc : + est L.C.I dans ℕ +</p> +</div> +</div> +<div id="outline-container-orgf679b38" class="outline-4"> +<h4 id="orgf679b38"><b>Example : soustraction</b></h4> +<div class="outline-text-4" id="text-orgf679b38"> +<p> +Est ce que la soustraction (-) est L.C.I dans ℕ? +</p> + +<p> +ℕ x ℕ —> ℕ +</p> + +<p> +(x,y) —> x - y ? +</p> + + +<p> +∃ x , y ∈ ℕ , x - y ∉ ℕ <i>En gros: il existe au moins une valeur de x et y dans ℕ tel que leur différence n’est <b>PAS</b> dans ℕ . tel que : si x est 5, et y c’est 9. Leur différence est -4, qui appartiens pas a ℕ</i> +</p> +</div> +</div> +</div> +<div id="outline-container-org5bd802e" class="outline-3"> +<h3 id="org5bd802e">La loi de composition externe dans E :</h3> +<div class="outline-text-3" id="text-org5bd802e"> +<p> +@ est L.C.E dans E, K est un corps +</p> + +<p> +K x E —> E +</p> + +<p> +(a,x) —> a @ x +</p> + +<p> +∀ (a , x) ∈ K x E , a @ x ∈ E +</p> +</div> +</div> +<div id="outline-container-orgbd9ed54" class="outline-3"> +<h3 id="orgbd9ed54">Groupes :</h3> +<div class="outline-text-3" id="text-orgbd9ed54"> +<p> +<i>Soit E un ensemble, soit @ une L.C.I dans E</i> +</p> + +<p> +(E, @) est un groupe Si : +</p> +</div> +<div id="outline-container-org8a61f65" class="outline-4"> +<h4 id="org8a61f65">Il contiens un élement neutre</h4> +<div class="outline-text-4" id="text-org8a61f65"> +<p> +∀ x ∈ E ; ∃ e ∈ E +</p> + +<p> +x @ e = e @ x = x +</p> + +<p> +On appelle <b>e</b> élement neutre +</p> + +<p> +<i>Ex: (ℕ,+) accepte un élement neutre, qui est 0, parceque x + 0 = 0 + x = x….cependent (ℕ,+) n’est pas un groupe. La raison est dans la prochaine condition</i> +</p> +</div> +</div> +<div id="outline-container-orgd7a2bc2" class="outline-4"> +<h4 id="orgd7a2bc2">Il contiens un élément symétrique</h4> +<div class="outline-text-4" id="text-orgd7a2bc2"> +<p> +∀ x ∈ E ; ∃ x’ ∈ E ; x @ x’ = x’ @ x = e +</p> + +<p> +On appelle <b>x’</b> élèment symétrique +</p> + +<p> +<i>Dans l’example en haut, on remarque qu’il n’y ya pas de chiffre x’ pour chaque chiffre x, qui est, l’hors de leur addition est egal a e (0), tout simplement car:</i> +</p> + +<p> +<i>x + x’ = e ; x + x’ = 0 ; x = -x’</i> +</p> + +<p> +<b>Or, Dans ℕ, on a pas de nombres négatifs</b> +</p> +</div> +</div> +<div id="outline-container-org942f964" class="outline-4"> +<h4 id="org942f964">@ est cummutative :</h4> +<div class="outline-text-4" id="text-org942f964"> +<p> +∀ (x , x’) ∈ E x E ; x @ x’ = x’ @ x +</p> + +<p> +<i>L’addition est cummutative, la soustraction ne l’es pas. 5 + 3 ou 3 + 5 est pareil, mais 5 - 3 et 3 - 5 sont différents</i> +</p> +</div> +</div> +</div> +<div id="outline-container-orgfd870b2" class="outline-3"> +<h3 id="orgfd870b2">Anneaux :</h3> +<div class="outline-text-3" id="text-orgfd870b2"> +<p> +Soit E un ensemble, (E , @ , !) est un anneau si : +</p> +</div> +<div id="outline-container-orgb630ae9" class="outline-4"> +<h4 id="orgb630ae9">(E ; @) est un groupe cummutatif</h4> +</div> +<div id="outline-container-org83da3f3" class="outline-4"> +<h4 id="org83da3f3">! est une loi associative :</h4> +<div class="outline-text-4" id="text-org83da3f3"> +<p> +∀ x , y , z ∈ E +</p> + +<p> +(x ! y) ! z = x ! (y ! z) +</p> +</div> +</div> +<div id="outline-container-org228d644" class="outline-4"> +<h4 id="org228d644">Distribution de ! par rapport à @ :</h4> +<div class="outline-text-4" id="text-org228d644"> +<p> +∀ x , y , z ∈ E +</p> + +<p> +(x @ y) ! z = ( x ! z ) @ ( y ! z ) +</p> +</div> +</div> +<div id="outline-container-org188625a" class="outline-4"> +<h4 id="org188625a">L’existance d’un élèment neutre de ! :</h4> +<div class="outline-text-4" id="text-org188625a"> +<p> +∀ x ∈ E , ∃ e ∈ E , x ! e = e ! x = x +</p> +</div> +</div> +<div id="outline-container-orge450b40" class="outline-4"> +<h4 id="orge450b40">! est cummutative :</h4> +<div class="outline-text-4" id="text-orge450b40"> +<p> +∀ x , y ∈ E , x ! y = y ! x +</p> +</div> +</div> +</div> +<div id="outline-container-orgb215ac1" class="outline-3"> +<h3 id="orgb215ac1">Corps :</h3> +<div class="outline-text-3" id="text-orgb215ac1"> +<p> +(E , @ , !) est un corps si les 5 conditions en haut sont vérifiées + cette condition : +</p> +</div> +<div id="outline-container-org51d906d" class="outline-4"> +<h4 id="org51d906d">La symétrie :</h4> +<div class="outline-text-4" id="text-org51d906d"> +<p> +∀ x ∈ E ; ∃ x’ ∈ E , x ! x’ = x’ ! x = e +</p> + +<p> +x’ est l’élément symétrique de x par rapport à ! +(sauf élément neutre première lois ) +</p> +</div> +</div> +</div> +<div id="outline-container-orgea20262" class="outline-3"> +<h3 id="orgea20262">Exercice : (ℝ, +, x) corps ou pas ?</h3> +<div class="outline-text-3" id="text-orgea20262"> +</div> +<div id="outline-container-org63b1ea5" class="outline-4"> +<h4 id="org63b1ea5">Est-ce un Anneau ?</h4> +<div class="outline-text-4" id="text-org63b1ea5"> +<ul class="org-ul"> +<li>(ℝ, +) est un groupe commutatif</li> +<li>x est une loi associative : (a x b) x c = a x (b x c)</li> +<li>On peut distribuer x par rapport a + : (a + b) x c = (a x c) + (b x c)</li> +<li>Il existe un élément neutre de x which is 1 : a x 1 = 1 x a = a</li> +<li>La multiplication est commutative : a x b = b x a</li> +</ul> + +<p> +Oui c’est un anneau +</p> +</div> +</div> +<div id="outline-container-org1c18b97" class="outline-4"> +<h4 id="org1c18b97">Est-ce un corps ?</h4> +<div class="outline-text-4" id="text-org1c18b97"> +<ul class="org-ul"> +<li>Oui : ∀ x ∈ ℝ\{e} ; x * x’ = 1</li> +</ul> +</div> +</div> +</div> +</div> +<div id="outline-container-org86b97eb" class="outline-2"> +<h2 id="org86b97eb">2nd cours :L’ordre dans ℝ, Majorant, minorant, borne superieure, borne inférieure <i>Oct 3</i> :</h2> +<div class="outline-text-2" id="text-org86b97eb"> +</div> +<div id="outline-container-org1f1c4d8" class="outline-3"> +<h3 id="org1f1c4d8">L’ordre dans ℝ</h3> +<div class="outline-text-3" id="text-org1f1c4d8"> +<p> +(ℝ, +, x) est un corps, Soit R une relation d’ordre dans ℝ si : +</p> + +<ol class="org-ol"> +<li><p> +R est antisymétrique : +</p> + +<p> +∀ x, y ℝ ; (x R y et y R x) ⇒ (x = y) +</p></li> + +<li><p> +R est reflexive : +</p> + +<p> +∀ x ∈ ℝ ; x R x +</p></li> + +<li>R est transitive : +∀ x, y, z ∈ ℝ , (x R y and y R z) ⇒ x R z</li> +</ol> +</div> +<div id="outline-container-orgc178857" class="outline-4"> +<h4 id="orgc178857">Exemples :</h4> +<div class="outline-text-4" id="text-orgc178857"> +</div> +<ul class="org-ul"> +<li><a id="org7b5f181"></a>Exemple numéro 1:<br /> +<div class="outline-text-5" id="text-org7b5f181"> +<p> +(ℝ , +, x) est un corps. Est ce la relation < est une relation d’ordre dans ℝ ? +</p> + + +<p> +Non, pourquoi ? parce que elle est pas réflexive : ∀ x ∈ ℝ, x < x <b><b>is obviously false</b></b> +</p> +</div> +</li> +<li><a id="org7f902ab"></a>Exemple numéro 2:<br /> +<div class="outline-text-5" id="text-org7f902ab"> +<p> +(ℝ , +, x) est un corps. Est ce la relation ≥ est une relation d’ordre dans ℝ ? +</p> + +<ol class="org-ol"> +<li>(Antisymétrique) ∀ x, y ℝ ; (x ≥ y AND y ≥ x) ⇒ x = y is true</li> +<li>(Réflexive) ∀ x, y ℝ ; x ≥ x is true</li> +<li>(Transitive) ∀ x, y, z ℝ ; (x ≥ y AND y ≥ z) ⇒ x ≥ z is also true</li> +</ol> +</div> +</li> +</ul> +</div> +</div> +<div id="outline-container-org446671f" class="outline-3"> +<h3 id="org446671f">Majorant, minorant, borne supérieure, borne inférieure</h3> +<div class="outline-text-3" id="text-org446671f"> +</div> +<div id="outline-container-org86c077d" class="outline-4"> +<h4 id="org86c077d">Majorant:</h4> +<div class="outline-text-4" id="text-org86c077d"> +<p> +Soit E un sous-ensemble de ℝ (E ⊆ ℝ) +</p> + + +<p> +Soit a ∈ ℝ, a est un majorant de E Si :∀ x ∈ E , x ≤ a +</p> +</div> +</div> +<div id="outline-container-org200a2c3" class="outline-4"> +<h4 id="org200a2c3">Minorant:</h4> +<div class="outline-text-4" id="text-org200a2c3"> +<p> +Soit E un sous-ensemble de ℝ (E ⊆ ℝ) +</p> + + +<p> +Soit b ∈ ℝ, b est un minorant de E Si :∀ x ∈ E , x ≥ b +</p> +</div> +</div> +<div id="outline-container-org46cb24b" class="outline-4"> +<h4 id="org46cb24b">Borne supérieure:</h4> +<div class="outline-text-4" id="text-org46cb24b"> +<p> +La borne supérieure est le plus petit des majorants <i>Sup(E) = Borne supérieure</i> +</p> +</div> +</div> +<div id="outline-container-orgbb1980a" class="outline-4"> +<h4 id="orgbb1980a">Borne inférieure:</h4> +<div class="outline-text-4" id="text-orgbb1980a"> +<p> +La borne inférieure est le plus grand des minorant <i>Inf(E) = Borne inférieure</i> +</p> +</div> +</div> +<div id="outline-container-org3159ba9" class="outline-4"> +<h4 id="org3159ba9">Maximum :</h4> +<div class="outline-text-4" id="text-org3159ba9"> +<p> +E ⊆ ℝ, a est un maximum de E (Max(E)) Si : a ∈ E ; ∀x ∈ E, x ≤ a. +</p> +</div> +</div> +<div id="outline-container-orgf43e25a" class="outline-4"> +<h4 id="orgf43e25a">Minimum :</h4> +<div class="outline-text-4" id="text-orgf43e25a"> +<p> +E ⊆ ℝ, b est un minimum de E (Min(E)) Si : b ∈ E ; ∀x ∈ E, x ≥ b. +</p> +</div> +</div> +<div id="outline-container-orga08da55" class="outline-4"> +<h4 id="orga08da55">Remarques :</h4> +<div class="outline-text-4" id="text-orga08da55"> +<p> +A et B deux ensembles bornés (Minoré et Majoré) : +</p> +<ol class="org-ol"> +<li>A ∪ B est borné</li> +<li>A ∩ B est borné</li> +<li>Sup(A ∪ B)= Max(sup A, sup B)</li> +<li>Inf(A ∩ B)= Min(inf A, inf B)</li> +<li>Sup(A ∩ B)= Min(sup A, sup B) <i>Le plus petit des Supérieur de A et B</i></li> +<li>Inf(A ∩ B)= Max(inf A, inf B) <i>Le plus grand des inférieur de A et B</i></li> +</ol> +</div> +</div> +</div> +</div> +<div id="outline-container-org05e73a3" class="outline-2"> +<h2 id="org05e73a3">3rd cours :Les suites numériques <i>Oct 5</i> :</h2> +<div class="outline-text-2" id="text-org05e73a3"> +</div> +<div id="outline-container-orgcd4347a" class="outline-4"> +<h4 id="orgcd4347a">Définition :</h4> +<div class="outline-text-4" id="text-orgcd4347a"> +<p> +Soit (Un)n ∈ ℕ une suite numérique , (Un)n est une application de ℕ dans ℝ: +</p> + + +<p> +ℕ -—> ℝ +</p> + + +<p> +n -—> U(n) = Un +</p> + +<ol class="org-ol"> +<li>(Un) ou (Un)n ∈ ℝ : une suite</li> +<li>Un : terme général</li> +</ol> +</div> +<ul class="org-ul"> +<li><a id="org495644f"></a>Exemple :<br /> +<div class="outline-text-6" id="text-org495644f"> +<p> +U : ℕ* -—> ℝ +</p> + + +<p> +n -—> 1/n +</p> + + +<p> +(Un) est une suite définit par Un = 1/n +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-orgdf8ea2e" class="outline-4"> +<h4 id="orgdf8ea2e">Définition N°2 :</h4> +<div class="outline-text-4" id="text-orgdf8ea2e"> +<p> +On peut définir une suite â partir d’une relation de récurrence entre deux termes successifs et le premier terme. +</p> +</div> +<ul class="org-ul"> +<li><a id="org0c7274e"></a>Exemple :<br /> +<div class="outline-text-6" id="text-org0c7274e"> +<p> +U(n+1) = Un /2 +</p> + + +<p> +U(1)= 1 +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-orgecfb02c" class="outline-3"> +<h3 id="orgecfb02c">Opérations sur les suites :</h3> +<div class="outline-text-3" id="text-orgecfb02c"> +</div> +<div id="outline-container-orgc56ccde" class="outline-4"> +<h4 id="orgc56ccde">La somme :</h4> +<div class="outline-text-4" id="text-orgc56ccde"> +<p> +Soient (Un) et (Vn) deux suites, la somme de (Un) et (Vn) est une suite de terme général Un + Vn +</p> +</div> +</div> +<div id="outline-container-orgfe0c8af" class="outline-4"> +<h4 id="orgfe0c8af">Le produit :</h4> +<div class="outline-text-4" id="text-orgfe0c8af"> +<p> +Soient (Un)n et (Vn)n deux suites alors (Un) x (Vn) est une autre suite de terme général Un x Vn +</p> +</div> +</div> +<div id="outline-container-orgfa9fe65" class="outline-4"> +<h4 id="orgfa9fe65">Inverse d’une suite :</h4> +<div class="outline-text-4" id="text-orgfa9fe65"> +<p> +Soit Un une suite de terme général Un alors l’inverse de (Un) est une autre suite (Vn) = 1/(Un) de terme général de Vn = 1/Un +</p> +</div> +</div> +<div id="outline-container-org09108ca" class="outline-4"> +<h4 id="org09108ca">Produit d’une suite par un scalaire :</h4> +<div class="outline-text-4" id="text-org09108ca"> +<p> +Soit (Un) une suite de T.G Un +</p> + + +<p> +∀ λ ∈ ℝ , λ(Un) n ∈ ℕ est une suite de T.G Vn= λUn +</p> +</div> +</div> +</div> +<div id="outline-container-org0d16671" class="outline-3"> +<h3 id="org0d16671">Suite bornée :</h3> +<div class="outline-text-3" id="text-org0d16671"> +<p> +Une suite (Un) est bornée si (Un) majorée et minorée +</p> +</div> +</div> +<div id="outline-container-org1819f18" class="outline-3"> +<h3 id="org1819f18">Suite majorée :</h3> +<div class="outline-text-3" id="text-org1819f18"> +<p> +Soit (Un) une suite +</p> + + +<p> +U : (Un) est majorée par M ∈ ℝ ; ∀ n ∈ ℕ ; ∃ M ∈ ℝ , Un ≤ M +</p> +</div> +</div> +<div id="outline-container-org3c0fbec" class="outline-3"> +<h3 id="org3c0fbec">Suite minorée :</h3> +<div class="outline-text-3" id="text-org3c0fbec"> +<p> +Soit (Un) une suite +</p> + + +<p> +U : (Un) est minorée par M ∈ ℝ ; ∀ n ∈ ℕ ; ∃ M ∈ ℝ , Un ≥ M +</p> +</div> +</div> +<div id="outline-container-orgebf3d3b" class="outline-3"> +<h3 id="orgebf3d3b">Suites monotones :</h3> +<div class="outline-text-3" id="text-orgebf3d3b"> +</div> +<div id="outline-container-org193a450" class="outline-4"> +<h4 id="org193a450">Les suites croissantes :</h4> +<div class="outline-text-4" id="text-org193a450"> +<p> +Soit (Un)n est une suite +</p> + + +<p> +(Un) est croissante si : ∀ n ∈ ℕ ; U(n+1) - Un ≥ 0 ⇔ Un+1 ≥ Un +</p> +</div> +</div> +<div id="outline-container-orgff47924" class="outline-4"> +<h4 id="orgff47924">Les suites décroissantes :</h4> +<div class="outline-text-4" id="text-orgff47924"> +<p> +Soit (Un)n est une suite +</p> + + +<p> +(Un) est décroissante si : ∀ n ∈ ℕ ; U(n+1) - Un ≤ 0 ⇔ Un+1 ≤ Un +</p> +</div> +</div> +</div> +</div> +<div id="outline-container-orgf08c70b" class="outline-2"> +<h2 id="orgf08c70b">Série TD N°1 : <i>Oct 6</i></h2> +<div class="outline-text-2" id="text-orgf08c70b"> +</div> +<div id="outline-container-org24ad469" class="outline-3"> +<h3 id="org24ad469">Exo 1 :</h3> +<div class="outline-text-3" id="text-org24ad469"> +</div> +<div id="outline-container-org542ddd3" class="outline-4"> +<h4 id="org542ddd3">Ensemble A :</h4> +<div class="outline-text-4" id="text-org542ddd3"> +<p> +A = {-1/n , n ∈ ℕ *} +</p> +</div> +<ul class="org-ul"> +<li><a id="org8b8384f"></a>Borne inférieure<br /> +<div class="outline-text-5" id="text-org8b8384f"> +<p> +∀ n ∈ ℕ* , -1/n ≥ -1 . -1 est la borne inférieure de l’ensemble A +</p> +</div> +</li> +<li><a id="orgd7452d9"></a>Minimum :<br /> +<div class="outline-text-5" id="text-orgd7452d9"> +<p> +∀ n ∈ ℕ* , -1/n ≥ -1 . -1 est le Minimum de l’ensemble A +</p> +</div> +</li> +<li><a id="org60371ac"></a>Borne supérieure :<br /> +<div class="outline-text-5" id="text-org60371ac"> +<p> +∀ n ∈ ℕ* , -1/n ≤ 0 . 0 est la borne supérieure de l’ensemble A +</p> +</div> +</li> +<li><a id="orge3e4c79"></a>Maximum :<br /> +<div class="outline-text-5" id="text-orge3e4c79"> +<p> +L’ensemble A n’as pas de maximum +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-org71dc196" class="outline-4"> +<h4 id="org71dc196">Ensemble B :</h4> +<div class="outline-text-4" id="text-org71dc196"> +<p> +B = [-1 , 3[ ∩ ℚ +</p> +</div> +<ul class="org-ul"> +<li><a id="orgb1a63f2"></a>Borne inférieure :<br /> +<div class="outline-text-5" id="text-orgb1a63f2"> +<p> +Inf(B) = Max(inf([-1 , 3[) , inf(ℚ)) +</p> + + +<p> +Puisse que ℚ n’as pas de Borne inférieure, donc par convention c’est <b>-∞</b>, +</p> + + +<p> +<b>Inf(B) = -1</b> +</p> +</div> +</li> +<li><a id="org777951c"></a>Borne supérieure :<br /> +<div class="outline-text-5" id="text-org777951c"> +<p> +Sup(B) = Min(sup([-1 ,3[) , sup(ℚ)) +</p> + + +<p> +Puisse que ℚ n’as pas de Borne supérieure, donc par convention c’est <b>+∞</b>, +</p> + + +<p> +<b>Sup(B) = 3</b> +</p> +</div> +</li> +<li><a id="org8f243eb"></a>Minimum :<br /> +<div class="outline-text-5" id="text-org8f243eb"> +<p> +<b>Min(B) = -1</b> +</p> +</div> +</li> +<li><a id="org7e50683"></a>Maximum :<br /> +<div class="outline-text-5" id="text-org7e50683"> +<p> +L’ensemble B n’as pas de Maximum +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-org22ac863" class="outline-4"> +<h4 id="org22ac863">Ensemble C :</h4> +<div class="outline-text-4" id="text-org22ac863"> +<p> +C = {3n ,n ∈ ℕ} +</p> +</div> +<ul class="org-ul"> +<li><a id="org21834f8"></a>Borne inférieure :<br /> +<div class="outline-text-5" id="text-org21834f8"> +<p> +Inf(C) = 0 +</p> +</div> +</li> +<li><a id="org9559808"></a>Borne supérieure :<br /> +<div class="outline-text-5" id="text-org9559808"> +<p> +Sup(C) = +∞ +</p> +</div> +</li> +<li><a id="org4a57e53"></a>Minimum :<br /> +<div class="outline-text-5" id="text-org4a57e53"> +<p> +Min(C) = 0 +</p> +</div> +</li> +<li><a id="org621b5ba"></a>Maximum :<br /> +<div class="outline-text-5" id="text-org621b5ba"> +<p> +L’ensemble C n’as pas de Maximum +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-orgfbcec21" class="outline-4"> +<h4 id="orgfbcec21">Ensemble D :</h4> +<div class="outline-text-4" id="text-orgfbcec21"> +<p> +D = {1 - 1/n , n ∈ ℕ*} +</p> +</div> +<ul class="org-ul"> +<li><a id="orga8dbb3d"></a>Borne inférieure :<br /> +<div class="outline-text-5" id="text-orga8dbb3d"> +<p> +Inf(D)= 0 +</p> +</div> +</li> +<li><a id="org17babcd"></a>Borne supérieure :<br /> +<div class="outline-text-5" id="text-org17babcd"> +<p> +Sup(D)= 1 +</p> +</div> +</li> +<li><a id="orgbd6de63"></a>Minimum :<br /> +<div class="outline-text-5" id="text-orgbd6de63"> +<p> +Min(D)= 0 +</p> +</div> +</li> +<li><a id="org8c6ce24"></a>Maximum :<br /> +<div class="outline-text-5" id="text-org8c6ce24"> +<p> +L’ensemble D n’as pas de Maximum +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-orgffb8405" class="outline-4"> +<h4 id="orgffb8405">Ensemble E :</h4> +<div class="outline-text-4" id="text-orgffb8405"> +<p> +E = { [2n + (-1)^n]/ n + 1 , n ∈ ℕ } +</p> + + +<p> +<b>Les valeurs que E peut prendre sont : “(2n + 1)/(n+1)” Si n est pair, et “(2n - 1)/(n+1)” si n est impair</b> +</p> + + +<p> +<b>On définit un ensemble F et G : F = { (2n + 1)/ (n+1) , n ∈ 2k}, G = { (2n - 1)/(n+1), n ∈ 2k+1}</b> +</p> + + +<p> +<b>Donc E = F ∪ G</b> +</p> +</div> +<ul class="org-ul"> +<li><a id="org600d86b"></a>Borne inférieure :<br /> +<div class="outline-text-5" id="text-org600d86b"> +<p> +Inf(E) = Min(inf(F), inf(G)) +</p> + + +<p> +Inf(F) = 1 ; Inf(G) = -1 +</p> + + +<p> +<b>Inf(E)= -1</b> +</p> +</div> +</li> +<li><a id="orgba6c1f0"></a>Borne supérieure :<br /> +<div class="outline-text-5" id="text-orgba6c1f0"> +<p> +Sup(E) = Max(sup(F), sup(G)) +</p> + + +<p> +sup(F) = +∞ ; sup(G) = +∞ +</p> + + +<p> +<b>Sup(E)= +∞</b> +</p> +</div> +</li> +<li><a id="orgc3c7881"></a>Minimum :<br /> +<div class="outline-text-5" id="text-orgc3c7881"> +<p> +Min(E)= -1 +</p> +</div> +</li> +<li><a id="org8d1ee35"></a>Maximum :<br /> +<div class="outline-text-5" id="text-org8d1ee35"> +<p> +E n’as pas de maximum +</p> +</div> +</li> +</ul> +</div> +</div> +<div id="outline-container-org738c8ed" class="outline-3"> +<h3 id="org738c8ed">Exo 2 :</h3> +<div class="outline-text-3" id="text-org738c8ed"> +</div> +<div id="outline-container-org3384f69" class="outline-4"> +<h4 id="org3384f69">Ensemble A :</h4> +<div class="outline-text-4" id="text-org3384f69"> +<p> +A = {x ∈ ℝ , 0 < x <√3} +</p> +</div> +<ul class="org-ul"> +<li><a id="orge3c7b72"></a>Borné<br /> +<div class="outline-text-5" id="text-orge3c7b72"> +<p> +<b>Oui</b>, Inf(A)= 0 ; Sup(A)=√3 +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-orgbbecb74" class="outline-4"> +<h4 id="orgbbecb74">Ensemble B :</h4> +<div class="outline-text-4" id="text-orgbbecb74"> +<p> +B = { x ∈ ℝ , 1/2 < sin x <√3/2} ; +</p> +</div> +<ul class="org-ul"> +<li><a id="org563c4c7"></a>Borné<br /> +<div class="outline-text-5" id="text-org563c4c7"> +<p> +<b>∀ x ∈ B, sin x > 1/2 ∴ Inf(B)= 1/2</b> +</p> + + +<p> +<b>∀ x ∈ B, sin x < √3/2 ∴ Sup(B)= √3/2</b> +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-org9718c4a" class="outline-4"> +<h4 id="org9718c4a">Ensemble C :</h4> +<div class="outline-text-4" id="text-org9718c4a"> +<p> +C = {x ∈ ℝ , x³ > 3} +</p> +</div> +<ul class="org-ul"> +<li><a id="org7b5f08b"></a>Minoré<br /> +<div class="outline-text-5" id="text-org7b5f08b"> +<p> +<b>∀ x ∈ C, x³ > 3 ∴ Inf(C)= 3</b> +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-org43e1433" class="outline-4"> +<h4 id="org43e1433">Ensemble D :</h4> +<div class="outline-text-4" id="text-org43e1433"> +<p> +D = {x ∈ ℝ , e^x < 1/2} +</p> +</div> +<ul class="org-ul"> +<li><a id="orga33ed32"></a>Borné<br /> +<div class="outline-text-5" id="text-orga33ed32"> +<p> +<b>∀ x ∈ C, e^x > 0 ∴ Inf(C)= 0</b> +</p> + + +<p> +<b>∀ x ∈ C, e^x < 1/2 ∴ Sup(C)= 1/2</b> +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-org968305a" class="outline-4"> +<h4 id="org968305a">Ensemble E :</h4> +<div class="outline-text-4" id="text-org968305a"> +<p> +E = {x ∈ ℝ , ∃ p ∈ ℕ* : x = √2/p} +</p> +</div> +<ul class="org-ul"> +<li><a id="org43c7a7b"></a>Majoré<br /> +<div class="outline-text-5" id="text-org43c7a7b"> +<p> +p = √2/x . Donc : <b>Sup(E)=1</b> +</p> +</div> +</li> +</ul> +</div> +</div> +<div id="outline-container-org034d6b3" class="outline-3"> +<h3 id="org034d6b3">Exo 3 :</h3> +<div class="outline-text-3" id="text-org034d6b3"> +<p> +U0 = 3/2 ; U(n+1) = (Un - 1)² + 1 +</p> +</div> +<div id="outline-container-org0e55fe7" class="outline-4"> +<h4 id="org0e55fe7">Question 1 :</h4> +<div class="outline-text-4" id="text-org0e55fe7"> +<p> +Montrer que : ∀ n ∈ ℕ , 1 < Un < 2 . +</p> + + +<p> +<b>(Un - 1)² ≥ 0 <i>Parce que c’est un carré</i></b> +</p> + + +<p> +<b>(Un - 1)² + 1 > 1</b> ; <b>U(n+1) ≥ 1</b> +</p> +</div> +<ul class="org-ul"> +<li><a id="org354cb33"></a>Raisonnement par récurrence :<br /> +<div class="outline-text-5" id="text-org354cb33"> +<p> +P(n) : ∀ n ∈ ℕ ; 1 < Un < 2 +</p> + + +<p> +P(0) est vraie : 1 < 3/2 < 2 +</p> + + +<p> +On suppose que P(n) est vraie et on vérifie P(n+1) pour une contradiction +</p> + + +<p> +1< Un < 2 ; 0 < Un - 1 < 1 ; 0 < (Un - 1)² < 1 ; 1 < (Un - 1)² + 1< 2 ; <b>1 < U(n+1) < 2</b> Donc elle est correcte +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-orgce11f0b" class="outline-4"> +<h4 id="orgce11f0b">Question 2 :</h4> +<div class="outline-text-4" id="text-orgce11f0b"> +<p> +Montrer que (Un)n est strictement monotone : +</p> + + +<p> +<b>U(n+1) - Un = (Un - 1)² + 1 - Un</b> ; <b>U(n+1) - Un = Un² + 1 - 2Un + 1 - Un</b> ; <b>U(n+1) - Un = Un² - 3Un + 2</b> +</p> + + +<p> +On étudie <b>Un² - 3Un + 2</b> sur l’intervalle ]1, 2[ : Un² - 3Un + 2 = 0 est une équation du 2nd ordre, <b>Δ = 1</b> , elle accepte deux solutions : Un = 1 et Un = 2 +</p> + + +<p> +On déduit que <b>Un² - 3Un + 2</b> est négatif sur [1 , 2] et positif en dehors, donc <b>∀ 1 < Un < 2 , Un² - 3Un + 2 < 0</b> ; <b>∀ 1 < Un < 2 , U(n+1) - Un < 0</b> ; <b>∀ 1 < Un < 2 , U(n+1) < Un</b> Donc (Un)n est une suite strictement monotonne décroissante +</p> +</div> +</div> +</div> +</div> +<div id="outline-container-org0b95370" class="outline-2"> +<h2 id="org0b95370">4th cours (Suite) : <i>Oct 10</i></h2> +<div class="outline-text-2" id="text-org0b95370"> +</div> +<div id="outline-container-org37a71e2" class="outline-3"> +<h3 id="org37a71e2">Les suites convergentes</h3> +<div class="outline-text-3" id="text-org37a71e2"> +<p> +Soit (Un)n est une suite convergente si lim Un n–> +∞ = l +</p> +</div> +<div id="outline-container-org4a6d045" class="outline-4"> +<h4 id="org4a6d045">Remarque :</h4> +<div class="outline-text-4" id="text-org4a6d045"> +<ol class="org-ol"> +<li>Un est une suite convergente alors Un est bornee</li> +<li>Un est une suite convergente lim Un n—> +∞ = l ⇔ lim |Un| n—> +∞ = |l|</li> +<li>Un est une suite majoree et croissante ⇒ Un converge</li> +<li>Un est une suite minoree et decroissante ⇒ Un converge</li> +<li>Soient (Un) et (Vn) deux suites convergentes, alors +<ol class="org-ol"> +<li>Un + Vn est convergente</li> +<li>Un * Vn est convergente</li> +<li>∀λ ∈ ℝ , (λUn) converge</li> +</ol></li> +<li>Soit Un est une suite bornee et soit Vn une suite. lim Vn n->+∞ = 0 Alors lim Vn * Un n-> +∞ = 0</li> +</ol> +</div> +</div> +</div> +<div id="outline-container-orgcd9ef0a" class="outline-3"> +<h3 id="orgcd9ef0a">Theoreme d’encadrement</h3> +<div class="outline-text-3" id="text-orgcd9ef0a"> +<p> +Soient Un Vn et Wn trois suites ∀n ∈ ℕ, Un ≤ Vn ≤ Wn . et lim Un n->∞ = lim Wn n-> +∞ = l ⇒ lim Vn n-> +∞ = l +</p> +</div> +</div> +</div> +</div> +<div id="postamble" class="status"> +<p class="author">Author: Crystal</p> +<p class="date">Created: 2023-10-11 Wed 19:18</p> +</div> +</body> +</html> \ No newline at end of file diff --git a/uni_notes/architecture.html b/uni_notes/architecture.html new file mode 100755 index 0000000..b5ba235 --- /dev/null +++ b/uni_notes/architecture.html @@ -0,0 +1,899 @@ +<?xml version="1.0" encoding="utf-8"?> +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" +"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> +<head> +<!-- 2023-10-11 Wed 19:53 --> +<meta http-equiv="Content-Type" content="text/html;charset=utf-8" /> +<meta name="viewport" content="width=device-width, initial-scale=1" /> +<title>Architecture 1</title> +<meta name="author" content="Crystal" /> +<meta name="generator" content="Org Mode" /> +<link rel="stylesheet" type="text/css" href="../src/css/colors.css"/> +<link rel="stylesheet" type="text/css" href="../src/css/style.css"/> +</head> +<body> +<div id="content" class="content"> +<h1 class="title">Architecture 1</h1> +<div id="outline-container-org644596f" class="outline-2"> +<h2 id="org644596f">Premier cours : Les systémes de numération <i>Sep 27</i> :</h2> +<div class="outline-text-2" id="text-org644596f"> +<p> +Un système de numération est une méthode pour représenter des nombres à l’aide de symboles et de règles. Chaque système, comme le décimal (base 10) ou le binaire (base 2), utilise une base définie pour représenter des valeurs numériques. Il est caractérisé par 3 entitiés mathématiques importantes: +</p> + +<ol class="org-ol"> +<li>Une base (genre 10, ou 2)</li> +<li>Un ensemble de chiffres</li> +<li>Des régles de représentations des nombres</li> +</ol> +</div> +<div id="outline-container-org93dcedb" class="outline-4"> +<h4 id="org93dcedb"><b>Examples :</b></h4> +<div class="outline-text-4" id="text-org93dcedb"> +<p> +<i>B10 est un systéme de numération caractérisé par:</i> +</p> +<ul class="org-ul"> +<li>Base = 10</li> +<li>Un ensemble de chiffres : (0,1,2,3,4,5,6,7,8,9)</li> +</ul> + +<p> +<i>B16 est un autre systéme de numération caractérisé par:</i> +</p> +<ul class="org-ul"> +<li>Base = 16</li> +<li><p> +Un ensemble de chiffres : (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F) +</p> + +<p> +<b>Puisse-qu’on peut pas utiliser des nombres a deux chiffres, on utilise des lettres aprés 9, en leur donnant des valeurs tel que :</b> +</p> + +<p> +A : 10 ; B : 11 ; C : 12 ; D : 13 ; E : 14 ; F : 15 +</p></li> +</ul> +</div> +</div> +<div id="outline-container-orgc27c000" class="outline-3"> +<h3 id="orgc27c000">Comment passer d’un systéme a base 10 a un autre</h3> +<div class="outline-text-3" id="text-orgc27c000"> +<p> +On symbolise un chiffre dans la base x par : (Nombre)x +</p> +</div> +<div id="outline-container-org54f4213" class="outline-4"> +<h4 id="org54f4213">Pour les chiffres entiers :</h4> +<div class="outline-text-4" id="text-org54f4213"> +<p> +<b>On fait une division successive, on prends le nombre 3257 comme exemple, on veut le faire passer d’une base décimale á une base 16:</b> +</p> + + +<p> +(3257)10 -—> (?)16 +</p> + + +<p> +On dévise 3257 par 16, et les restants de la division serra la valeur en base16: +</p> + +<p> +3257/16 = 203 + <b>9</b> / 16 +</p> + +<p> +203/16 = 12 + <b>B</b> / 16 <i>REMARQUE, 11 N’APPARTIENS PAS A L’ENSEMBLE DES CHIFFRES EN BASE16, CE QUI VEUT DIRE QU’ON LE REMPLACE PAR SON EQUIVALENT, DANS CE CAS LA: <b>B</b></i> +</p> + +<p> +12/16 = 0 + <b>C</b> / 16 <i>Pareil ici, 12 n’existe pas, donc c’est C. Autre note : La division s’arréte quand le résultat de la division est nul</i> +</p> +</div> +<ul class="org-ul"> +<li><a id="orgc41ba5c"></a><b>Conclusion:</b><br /> +<div class="outline-text-5" id="text-orgc41ba5c"> +<p> +(3257)10 -—> (CB9)16 +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-org6f8ff12" class="outline-4"> +<h4 id="org6f8ff12">Pour les chiffres non entiers :</h4> +<div class="outline-text-4" id="text-org6f8ff12"> +<p> +<b>On fait la division successive pour la partie entiére, et une multiplication successive pour la partie rationelle:</b> +</p> + +<p> +(3257,32)10 -—> (?)16 +</p> + +<p> +On a déja la partie entiére donc on s’occupe de la partie aprés la virgule: +</p> + +<p> +0,32 x 16 = <b>5</b>,12 +</p> + +<p> +0,12 x 16 = <b>1</b>,92 +</p> + +<p> +0,92 x 16 = <b>E</b>,72 <i>On a pas de 15 donc c’est un E</i> +</p> + +<p> +0,72 x 16 = <b>B</b>,52 +</p> + +<p> +0,52 x 16 = <b>8</b>,32 +</p> + +<p> +0,32 x 16 = <b>5</b>,12 +</p> + +<p> +… +</p> + +<p> +<i>On s’arréte quand on trouve un chiffre entier, et si on trouve pas, on s’arréte quand on remarque une répetition, dans ce cas la, la séquance 51EB8 vas se répéter indéfiniment, donc on se contente d’écrire la partie qui se répéte avec une barre en haut</i> +</p> + + +<p> +(3257,32)10 -—> (CB9, <span class="underline">51EB8</span>)16 +</p> +</div> +</div> +</div> +</div> +<div id="outline-container-orga79d939" class="outline-2"> +<h2 id="orga79d939">2nd cours : Les systèmes de numération (Suite) <i>Oct 3</i> :</h2> +<div class="outline-text-2" id="text-orga79d939"> +</div> +<div id="outline-container-org3e165a8" class="outline-3"> +<h3 id="org3e165a8">Comment passer d’une base N a la base 10 :</h3> +<div class="outline-text-3" id="text-org3e165a8"> +<p> +Prenons comme exemple le nombre (11210,0011)3 , chaque chiffre dans ce nombre a un rang qui commence par 0 au premier chiffre (a gauche de la virgule) et qui augmente d’un plus qu’on avance a gauche, et diminue si on part a droite. Dans ce cas la : +</p> + + +<p> +(11210,0011)3 ; le 0 est de rang 0, le 1 est de rang 1, le 2 est de rang 2, le 1 est de rang 3, le 1 est de rang 4. Et si on part du coté de la virgule, 0 est de rang -1, 0 est de rang -2, le 1 est de rang -3, et le 1 est de rang -4. +</p> + + +<p> +Et pour passer a la base 10, il suffit d’appliquer cette formule : <b>Chiffre x Base^(rang) + 2emeChiffre x Base^(rang)… etc</b>, donc dans notre example: +</p> + + +<p> +<i>0 x 3° + 1 x 3¹ + 2 x 3² + 1 x 3³ + 1 x 3^4 + 0 x 3¯¹ + 0 x 3¯² + 1 x 3¯³ + 1 x 3^(-4) ≈ (129,05)10</i> +</p> +</div> +</div> +<div id="outline-container-org5a3bcfe" class="outline-3"> +<h3 id="org5a3bcfe">Comment passer d’une base N a une base N^(n) :</h3> +<div class="outline-text-3" id="text-org5a3bcfe"> +<p> +Si il ya une relation entre une base et une autre, on peut directement transformer vers cette base. +</p> +</div> +<div id="outline-container-orgab808d0" class="outline-4"> +<h4 id="orgab808d0">Exemple :</h4> +<div class="outline-text-4" id="text-orgab808d0"> +<p> +Pour passer de la base 2 a la base 8 (8 qui est 2³) on découpe les chiffres 3 par 3 +</p> + + +<p> +(1 101 011, 011)2 ; Pour le dernier 1 qui est seul <code>tout comme moi</code> il suffit d’ajouter des 0 à gauche (car on peut) pour compléter le découpage. +</p> + + +<p> +(001 101 011, 011)2; Next step c’est de dessiner le tableau de conversion de la base 2 a la base 8 ( un tableau a 3 bits ) +</p> + + +<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> + + +<colgroup> +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> + +<col class="org-right" /> +</colgroup> +<thead> +<tr> +<th scope="col" class="org-right">N</th> +<th scope="col" class="org-right"> </th> +<th scope="col" class="org-right"> </th> +<th scope="col" class="org-right"> </th> +</tr> +</thead> +<tbody> +<tr> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +</tr> + +<tr> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +</tr> + +<tr> +<td class="org-right">2</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +</tr> + +<tr> +<td class="org-right">3</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +</tr> + +<tr> +<td class="org-right">4</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">0</td> +</tr> + +<tr> +<td class="org-right">5</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +<td class="org-right">1</td> +</tr> + +<tr> +<td class="org-right">6</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">0</td> +</tr> + +<tr> +<td class="org-right">7</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +<td class="org-right">1</td> +</tr> +</tbody> +</table> + + +<p> +Pour remplir on a qu’a diviser les chiffres en deux, et mettre des 0 dans la première partie et des 1 dans la 2éme, et en faire de même pour les autres colonnes . +</p> + + +<p> +Maintenant il suffit de trouver l’équivalent de la base2 en base8 : +</p> + + +<p> +001 c’est 1 ; 101 c’est 5 ; 011 c’est 3 ; donc <b>(1101011,011)2 —> (153,3)8</b> +</p> +</div> +</div> +</div> +<div id="outline-container-orgbf931c5" class="outline-3"> +<h3 id="orgbf931c5">L’arithmétique binaire :</h3> +<div class="outline-text-3" id="text-orgbf931c5"> +</div> +<div id="outline-container-orgbe5bb2f" class="outline-4"> +<h4 id="orgbe5bb2f">L’addition :</h4> +<div class="outline-text-4" id="text-orgbe5bb2f"> +<p> +0 + 0 = 0 On retiens 0 +</p> + + +<p> +1 + 0 = 1 On retiens 0 +</p> + + +<p> +0 + 1 = 1 On retiens 0 +</p> + + +<p> +1 + 1 = 0 On retiens 1 +</p> + + +<p> +1 + 1 + 1 = 1 On retiens 1 +</p> + + +<p> +Donc 0110 + 1101 = 10011 +</p> +</div> +</div> +<div id="outline-container-org76fafea" class="outline-4"> +<h4 id="org76fafea">La soustraction :</h4> +<div class="outline-text-4" id="text-org76fafea"> +<p> +0 - 0 = 0 On emprunt = 0 +</p> + + +<p> +1 - 0 = 1 On emprunt = 0 +</p> + + +<p> +0 - 1 = 1 On emprunt = 1 +</p> + + +<p> +1 - 1 = 0 On emprunt = 0 +</p> +</div> +</div> +</div> +<div id="outline-container-org0db7791" class="outline-3"> +<h3 id="org0db7791">TP N°1 :</h3> +<div class="outline-text-3" id="text-org0db7791"> +</div> +<div id="outline-container-org602a6f4" class="outline-4"> +<h4 id="org602a6f4">Exo1:</h4> +<div class="outline-text-4" id="text-org602a6f4"> +<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> + + +<colgroup> +<col class="org-left" /> + +<col class="org-left" /> + +<col class="org-left" /> + +<col class="org-left" /> + +<col class="org-left" /> +</colgroup> +<thead> +<tr> +<th scope="col" class="org-left">Base 10</th> +<th scope="col" class="org-left">Base 2</th> +<th scope="col" class="org-left">Base 3</th> +<th scope="col" class="org-left">Base 8</th> +<th scope="col" class="org-left">Base 16</th> +</tr> +</thead> +<tbody> +<tr> +<td class="org-left">22,75</td> +<td class="org-left">10110,11</td> +<td class="org-left">211, <span class="underline">20</span></td> +<td class="org-left">26,6</td> +<td class="org-left">F6,C</td> +</tr> + +<tr> +<td class="org-left">684,125</td> +<td class="org-left">1010101100,001</td> +<td class="org-left">221100, <span class="underline">01</span></td> +<td class="org-left">1254,1</td> +<td class="org-left">2AC,2</td> +</tr> + +<tr> +<td class="org-left">3931,625</td> +<td class="org-left">111101011011,101</td> +<td class="org-left">1101121, <span class="underline">12</span></td> +<td class="org-left">7533,5</td> +<td class="org-left">F5B,A</td> +</tr> + +<tr> +<td class="org-left">52,38</td> +<td class="org-left">110100,011</td> +<td class="org-left">1221,101</td> +<td class="org-left">64,3</td> +<td class="org-left">34,6147</td> +</tr> + +<tr> +<td class="org-left">10,67</td> +<td class="org-left">1010,101</td> +<td class="org-left">23,5</td> +<td class="org-left">12,5</td> +<td class="org-left">A,AB85</td> +</tr> +</tbody> +</table> +</div> +<ul class="org-ul"> +<li><a id="org44fc05c"></a>(10110,11)2<br /> +<div class="outline-text-5" id="text-org44fc05c"> +<p> +0 x 2° + 1 x 2¹ + 1 x 2² + 0 x 2³ + 1 x 2^(4) + 1 x 2¯¹ + 1 x 2¯² = (22.75)10 +</p> +</div> +<ul class="org-ul"> +<li><a id="org6e587a2"></a>(22,75)10 -—> (3)<br /> +<div class="outline-text-6" id="text-org6e587a2"> +<p> +22/3 = 7 R <b>1</b> ; 7/3 = 2 R <b>1</b> ; 2/3 = 0 R <b>2</b> +</p> + + +<p> +0,75 x 3 = <b>2</b>.25 ; 0,25 x 3 = <b>0</b>.75 ….. +</p> + + +<p> +(22,75)10 -—> (211, <span class="underline">20</span>) +</p> +</div> +</li> +<li><a id="org5ed87ae"></a>(10110,11)2 -—> (8)<br /> +<div class="outline-text-6" id="text-org5ed87ae"> +<p> +8 = 2³ ; (010 110,110)2 -—> (?)8 +</p> + + +<p> +En utilisant le tableau 3bits : +</p> + + +<p> +010 : 2 ; 110 : 6 ; 110 : 6 +</p> + + +<p> +(10110,11)2 -—> (26,6)8 +</p> +</div> +</li> +<li><a id="org941eaad"></a>(22,75)10 -—> (16)<br /> +<div class="outline-text-6" id="text-org941eaad"> +<p> +22/16 = 1 R <b>6</b> ; 1/16 : 0 R <b>F</b> +</p> + + +<p> +0,75 x 16 = <b>C</b> +</p> + + +<p> +(22,75)10 -—> (F6,C)16 +</p> +</div> +</li> +</ul> +</li> +<li><a id="org14d08d6"></a>(1254,1)8<br /> +<div class="outline-text-5" id="text-org14d08d6"> +<p> +4 x 8° + 5 x 8¹ + 2 x 8² + 1 x 8³ + 1 x 8¯¹ = (684,125)10 +</p> +</div> +<ul class="org-ul"> +<li><a id="orgd83dbd8"></a>(1254,1)8 -—> (?)2<br /> +<div class="outline-text-6" id="text-orgd83dbd8"> +<p> +En utilisant le tableau 3bits : +</p> + + +<p> +001 010 101 100,001 <i>We will get rid of the leading zeros</i> +</p> + + +<p> +(1010101100,001)2 +</p> +</div> +</li> +<li><a id="org1897d7c"></a>(684,125)10 -—> (?)3<br /> +<div class="outline-text-6" id="text-org1897d7c"> +<p> +684/3 = 228 R <b>0</b> ; 228/3 = 76 R <b>0</b> ; 76/3 = 25 R <b>1</b> ; 25/3 = 8 R <b>1</b> ; 8/3 = 2 R <b>2</b> ; 2/3 = 0 R <b>2</b> +</p> + + +<p> +0,125 x 3 = <b>0</b>,375 ; 0,375 x 3 = <b>1</b>,125 +</p> + + +<p> +(221100, <span class="underline">01</span>)3 +</p> +</div> +</li> +<li><a id="org5d6fc0d"></a>(684,125)10 -—> (?)16<br /> +<div class="outline-text-6" id="text-org5d6fc0d"> +<p> +684/16 = 42 R <b>C</b> ; 42/16 = 2 R <b>A</b> ; 2/16 0 R <b>2</b> +</p> + + +<p> +0,125 x 16 = <b>2</b> +</p> + + +<p> +(2AC,2)16 +</p> +</div> +</li> +</ul> +</li> +<li><a id="orgdb748f1"></a>(F5B,A)16<br /> +<div class="outline-text-5" id="text-orgdb748f1"> +<p> +11 x 16° + 5 x 16 + 15 x 16² + 10 x 16¯¹ = (3931,625)10 +</p> +</div> +<ul class="org-ul"> +<li><a id="org68f19f3"></a>(3931,625)10 -—> (8)<br /> +<div class="outline-text-6" id="text-org68f19f3"> +<p> +3931/8 = 491 R <b>3</b> ; 491/8 = 61 R <b>3</b> ; 61/8 = 7 R <b>5</b> ; 7/8 = 0 R <b>7</b> +</p> + + +<p> +0,625 x 8 = <b>5</b> +</p> + + +<p> +(7533,5)8 +</p> +</div> +</li> +<li><a id="org39a13c6"></a>(7533,5)8 -—> (2)<br /> +<div class="outline-text-6" id="text-org39a13c6"> +<p> +En utilisant le tableau 3bits +</p> + +<p> +(111 101 011 011,101)2 +</p> +</div> +</li> +<li><a id="org8613276"></a>(3931,625)10 -—> (3)<br /> +<div class="outline-text-6" id="text-org8613276"> +<p> +3931/3 = 1310 R <b>1</b> ; 1310/3 = 436 R <b>2</b> ; 436/3 = 145 R <b>1</b> ; 145/3 = 48 R <b>1</b> ; 48/3 = 16 R <b>0</b> ; 16/3 = 5 R <b>1</b> ; 5/3 = 1 R <b>2</b> ; 1/3 = 0 R <b>1</b> +</p> + + +<p> +0.625 x 3 = <b>1</b>,875 ; 0,875 x 3 = <b>2</b>,625 +</p> + + +<p> +(1101121, <span class="underline">12</span>)3 +</p> +</div> +</li> +</ul> +</li> +<li><a id="orgf5ad2f0"></a>(52,38)10<br /> +<div class="outline-text-5" id="text-orgf5ad2f0"> +<p> +52/2 = 26 R <b>0</b> ; 26/2 = 13 R <b>0</b> ; 13/2 = 6 R <b>1</b> ; 6/2 = 3 R <b>0</b> ; 3/2 = 1 R <b>1</b> ; 1/2 = 0 R <b>1</b> +</p> + + +<p> +0,38 x 2 = <b>0</b>,76 ; 0,76 x 2 = <b>1</b>,52 ; 0,52 x 2 = <b>1</b>,04 ; 0,04 x 2 = <b>0</b>,08 …. +</p> + + +<p> +(110100,0110)2 +</p> +</div> +<ul class="org-ul"> +<li><a id="org40d1958"></a>(52,38)10 -—> (3)<br /> +<div class="outline-text-6" id="text-org40d1958"> +<p> +52/3 = 17 R <b>1</b> ; 17/3 = 5 R <b>2</b> ; 5/3 = 1 R <b>2</b> ; 1/3 = 0 R <b>1</b> +</p> + + +<p> +0,38 x 3 = <b>1</b>.14 ; 0,14 x 3 = <b>0</b>.42 ; 0,42 x 3 = <b>1</b>.26 ; 0.26 x 3 = <b>0</b>.78 … +</p> + + +<p> +(1221,101)3 +</p> +</div> +</li> +<li><a id="org15b5aba"></a>(110100,011)2 -—> (8)<br /> +<div class="outline-text-6" id="text-org15b5aba"> +<p> +En utilisant le tableau 3bits: +</p> + + +<p> +(110 100,011)2 -—> (64,3)8 +</p> +</div> +</li> +<li><a id="org495819e"></a>(52,38)10 -—> (16)<br /> +<div class="outline-text-6" id="text-org495819e"> +<p> +52/16 = 3 R <b>4</b> ; 3/16 = 0 R <b>3</b> +</p> + + +<p> +0,38 x 16 = <b>6</b>,08 ; 0,08 x 16 = <b>1</b>,28 ; 0,28 x 16 = <b>4</b>,48 ; 0,48 x 16 = <b>7</b>,68 …. +</p> + + +<p> +(34,6147)16 +</p> +</div> +</li> +</ul> +</li> +<li><a id="orga49bba8"></a>(23,5)3<br /> +<div class="outline-text-5" id="text-orga49bba8"> +<p> +3 x 3° + 2 x 3 + 5 x 3¯¹ = (10.67)10 +</p> +</div> +<ul class="org-ul"> +<li><a id="org2b7d27e"></a>(10,67)10 -—> (2)<br /> +<div class="outline-text-6" id="text-org2b7d27e"> +<p> +10/2 = 5 R <b>0</b> ; 5/2 = 2 R <b>1</b> ; 2/2 = 1 R <b>0</b> ; 1/2 = 0 R <b>1</b> +</p> + + +<p> +0,67 x 2 = <b>1</b>,34 ; 0,34 x 2 = <b>0</b>,68 ; 0,68 x 2 = <b>1</b>,36 ; 0,36 x 2 = <b>0</b>,72 … +</p> + + +<p> +(1010,101)2 +</p> +</div> +</li> +<li><a id="org7aad862"></a>(001 010,101)2 -—> (8)<br /> +<div class="outline-text-6" id="text-org7aad862"> +<p> +<b>Ô Magic 3bits table, save me soul, me children and me maiden:</b> +</p> + + +<p> +(12,5)8 +</p> +</div> +</li> +<li><a id="orgdb90500"></a>(10,67)10 -—> (16)<br /> +<div class="outline-text-6" id="text-orgdb90500"> +<p> +10/16 = 0 R <b>A</b> +</p> + + +<p> +0,67 x 16 = <b>A</b>,72 ; 0,72 x 16 = <b>B</b>,52 ; 0,52 x 16 = <b>8</b>,32 ; 0,32 x 16 = <b>5</b>,12 … +</p> + + +<p> +(A,AB85)16 +</p> +</div> +</li> +</ul> +</li> +</ul> +</div> +<div id="outline-container-orgd9aad1c" class="outline-4"> +<h4 id="orgd9aad1c">Exo2:</h4> +<div class="outline-text-4" id="text-orgd9aad1c"> +</div> +<ul class="org-ul"> +<li><a id="org9a44306"></a>(34)? = (22)10<br /> +<div class="outline-text-5" id="text-org9a44306"> +<p> +(34)a = (22)10 ; 4 x a° + 3 x a = 22 ; 4 + 3a = 22 ; 3a = 18 +</p> + + +<p> +<b>a = 6</b> +</p> +</div> +</li> +<li><a id="org621cd70"></a>(75)? = (117)10<br /> +<div class="outline-text-5" id="text-org621cd70"> +<p> +(75)b = (117)10 ; 5 x b° + 7 x b¹ = 117 ; 5 + 7b = 117 ; 7b = 112 +</p> + + +<p> +<b>b = 16</b> +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-org25943b2" class="outline-4"> +<h4 id="org25943b2">Exo3:</h4> +<div class="outline-text-4" id="text-org25943b2"> +</div> +<ul class="org-ul"> +<li><a id="orge98e1b3"></a>(101011)2 + (111011)2<br /> +<div class="outline-text-5" id="text-orge98e1b3"> +<p> +101011 + 111011 = 1100110 +</p> +</div> +</li> +<li><a id="org9336956"></a>(1011,1101)2 + (11,1)2<br /> +<div class="outline-text-5" id="text-org9336956"> +<p> +1011,1101 + 11,1000 = 1111,0101 +</p> +</div> +</li> +<li><a id="org8a605df"></a>(1010,0101)2 - (110,1001)2<br /> +<div class="outline-text-5" id="text-org8a605df"> +<p> +1010,0101 - 110,1001 = 11,1100 +</p> +</div> +</li> +</ul> +</div> +</div> +<div id="outline-container-orgd1bb996" class="outline-3"> +<h3 id="orgd1bb996">L’arithmétique binaire (Suite): <i>Oct 4</i></h3> +<div class="outline-text-3" id="text-orgd1bb996"> +</div> +<div id="outline-container-org1a9b0f2" class="outline-4"> +<h4 id="org1a9b0f2">La multiplication :</h4> +<div class="outline-text-4" id="text-org1a9b0f2"> +<p> +0 x 0 = 0 +</p> + + +<p> +0 x 1 = 0 +</p> + + +<p> +1 x 0 = 0 +</p> + + +<p> +1 x 1 = 1 +</p> +</div> +</div> +<div id="outline-container-org17f1eab" class="outline-4"> +<h4 id="org17f1eab">La division :</h4> +<div class="outline-text-4" id="text-org17f1eab"> +<p> +On divise de la manière la plus normale du monde !!! +</p> +</div> +</div> +</div> +</div> +<div id="outline-container-org4e457be" class="outline-2"> +<h2 id="org4e457be">4th cours : Le codage <i>Oct 10</i></h2> +<div class="outline-text-2" id="text-org4e457be"> +</div> +<div id="outline-container-org8c7cda2" class="outline-3"> +<h3 id="org8c7cda2">Le codage des entiers positifs</h3> +<div class="outline-text-3" id="text-org8c7cda2"> +<p> +Le codage sur n bits permet de representer tout les entiers naturels compris entre [0, 2^n - 1]. On peut coder sur 8bits les entiers entre [0;2^8 - 1(255)] +</p> +</div> +</div> +<div id="outline-container-org1aa2cfc" class="outline-3"> +<h3 id="org1aa2cfc">Le codage des nombres relatifs</h3> +<div class="outline-text-3" id="text-org1aa2cfc"> +</div> +<div id="outline-container-orgc29141b" class="outline-4"> +<h4 id="orgc29141b">Remarque</h4> +<div class="outline-text-4" id="text-orgc29141b"> +<p> +Quelque soit le codage utilise, par convention le dernier bit est reserve pour le signe. ou 1 est negatif et 0 est positif. +</p> +</div> +</div> +<div id="outline-container-org4899ed4" class="outline-4"> +<h4 id="org4899ed4">Le codage en signe + valeur absolue (SVA):</h4> +<div class="outline-text-4" id="text-org4899ed4"> +<p> +Avec n bits le n eme est reserve au signe : [-(2^n-1)-1 , 2^n-1 -1]. Sur 8bits [-127, 127] +</p> +</div> +</div> +<div id="outline-container-orgc410845" class="outline-4"> +<h4 id="orgc410845">Codage en compliment a 1 (CR):</h4> +<div class="outline-text-4" id="text-orgc410845"> +<p> +On obtiens le compliment a 1 d’un nombre binaire en inversant chaqu’un de ses bits (1 -> 0 et 0-> 1) les nombres positifs sont la meme que SVA (il reste tel qu’il est) +</p> +</div> +</div> +<div id="outline-container-orgb19633c" class="outline-4"> +<h4 id="orgb19633c">Codage en compliment a 2 (CV):</h4> +<div class="outline-text-4" id="text-orgb19633c"> +<p> +C’est literallement CR + 1 pour les negatifs et SVA pour les nombres positifs +</p> +</div> +</div> +</div> +</div> +</div> +<div id="postamble" class="status"> +<p class="author">Author: Crystal</p> +<p class="date">Created: 2023-10-11 Wed 19:53</p> +</div> +</body> +</html> \ No newline at end of file |