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| author | Crystal <crystal@wizard.tower> | 2023-10-20 18:06:12 +0100 |
|---|---|---|
| committer | Crystal <crystal@wizard.tower> | 2023-10-20 18:06:12 +0100 |
| commit | 452f36f66cf56bd8b92677d6b6bbfb69ce54cfe4 (patch) | |
| tree | 585d0d333f391b645e99dfa9314b0309c1d64d10 /uni_notes/algebra.html | |
| parent | 3b1929a28bf43fe486ed5b9ba4d2a07adcb34b5f (diff) | |
| download | www-452f36f66cf56bd8b92677d6b6bbfb69ce54cfe4.tar.gz | |
Finally, an update
Diffstat (limited to 'uni_notes/algebra.html')
| -rwxr-xr-x | uni_notes/algebra.html | 1110 |
1 files changed, 640 insertions, 470 deletions
diff --git a/uni_notes/algebra.html b/uni_notes/algebra.html index 2129e72..fd6602c 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> -<!-- 2023-10-17 Tue 22:32 --> +<!-- 2023-10-20 Fri 15:12 --> <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> @@ -11,124 +11,233 @@ <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"> +<div id="org-div-home-and-up"> + <a accesskey="h" href="../../../uni_notes/"> UP </a> + | + <a accesskey="H" href="https://crystal.tilde.institute/"> HOME </a> +</div><div id="content" class="content"> <h1 class="title">Algebra 1</h1> -<div id="outline-container-org76eaba1" class="outline-2"> -<h2 id="org76eaba1">Contenu de la Matiére</h2> -<div class="outline-text-2" id="text-org76eaba1"> +<div id="table-of-contents" role="doc-toc"> +<h2>Table of Contents</h2> +<div id="text-table-of-contents" role="doc-toc"> +<ul> +<li><a href="#orgdb6601b">Contenu de la Matiére</a> +<ul> +<li><a href="#orgc89165c">Rappels et compléments (11H)</a></li> +<li><a href="#org1ae59da">Structures Algébriques (11H)</a></li> +<li><a href="#orgdcb51e0">Polynômes et fractions rationnelles</a></li> +</ul> +</li> +<li><a href="#org8356dfb">Premier cours : Logique mathématique et méthodes du raisonnement mathématique <i>Sep 25</i> :</a> +<ul> +<li><a href="#org4521340">Properties:</a> +<ul> +<li><a href="#org84e1ec3"><b>Absorption</b>:</a></li> +<li><a href="#org2345992"><b>Commutativity</b>:</a></li> +<li><a href="#orgfee557e"><b>Associativity</b>:</a></li> +<li><a href="#orgb55cfb2"><b>Distributivity</b>:</a></li> +<li><a href="#orgb0de2bb"><b>Neutral element</b>:</a></li> +<li><a href="#orgb51e625"><b>Negation of a conjunction & a disjunction</b>:</a></li> +<li><a href="#org4e29124"><b>Transitivity</b>:</a></li> +<li><a href="#org520b7b0"><b>Contraposition</b>:</a></li> +<li><a href="#org8159635">God only knows what this property is called:</a></li> +</ul> +</li> +<li><a href="#org4210e18">Some exercices I found online :</a> +<ul> +<li><a href="#org46d1ee6">USTHB 2022/2023 Section B :</a></li> +</ul> +</li> +</ul> +</li> +<li><a href="#org732b9dd">2éme cours <i>Oct 2</i></a> +<ul> +<li><a href="#orgdfba00a">Quantifiers</a> +<ul> +<li><a href="#org93d5891">Proprieties</a></li> +</ul> +</li> +<li><a href="#orgf21a239">Multi-parameter proprieties :</a></li> +<li><a href="#org779f917">Methods of mathematical reasoning :</a> +<ul> +<li><a href="#orgc6832c3">Direct reasoning :</a></li> +<li><a href="#orgf140b16">Reasoning by the Absurd:</a></li> +<li><a href="#org320dc57">Reasoning by contraposition:</a></li> +<li><a href="#org27943b2">Reasoning by counter example:</a></li> +</ul> +</li> +</ul> +</li> +<li><a href="#orgfddd579">3eme Cours : <i>Oct 9</i></a> +<ul> +<li> +<ul> +<li><a href="#org21eab57">Reasoning by recurrence :</a></li> +</ul> +</li> +</ul> +</li> +<li><a href="#org155c9a9">4eme Cours : Chapitre 2 : Sets and Operations</a> +<ul> +<li><a href="#org5bb0b39">Definition of a set :</a></li> +<li><a href="#org469078a">Belonging, inclusion, and equality :</a></li> +<li><a href="#org1461e60">Intersections and reunions :</a> +<ul> +<li><a href="#orgd9db499">Intersection:</a></li> +<li><a href="#org0946751">Union:</a></li> +<li><a href="#org8fb5e39">Difference between two sets:</a></li> +<li><a href="#org17ec98b">Complimentary set:</a></li> +<li><a href="#org61d7d25">Symentrical difference</a></li> +</ul> +</li> +<li><a href="#org1e0c21a">Proprieties :</a> +<ul> +<li><a href="#orgaed74b7">Commutativity:</a></li> +<li><a href="#org1d74822">Associativity:</a></li> +<li><a href="#org667f4dd">Distributivity:</a></li> +<li><a href="#org31c4f57">Lois de Morgan:</a></li> +<li><a href="#orgc30ba1d">An other one:</a></li> +<li><a href="#orgac52c86">An other one:</a></li> +<li><a href="#orge0de23e">And an other one:</a></li> +<li><a href="#org8277b0b">And the last one:</a></li> +</ul> +</li> +</ul> +</li> +<li><a href="#orgeb2675e">5eme cours: L’ensemble des parties d’un ensemble <i>Oct 16</i></a> +<ul> +<li> +<ul> +<li><a href="#org2c7d514">Notes :</a></li> +<li><a href="#org76da8f4">Examples :</a></li> +</ul> +</li> +<li><a href="#org3da3de1">Partition of a set :</a></li> +<li><a href="#org077b994">Cartesian products :</a> +<ul> +<li><a href="#org72822f5">Example :</a></li> +<li><a href="#org04e1be3">Some proprieties:</a></li> +</ul> +</li> +</ul> +</li> +<li><a href="#org416c1e1">Binary relations in a set :</a> +<ul> +<li><a href="#org5ea795f">Definition :</a></li> +<li><a href="#orgb9d678f">Proprieties :</a></li> +<li><a href="#org6df8952">Equivalence relationship :</a> +<ul> +<li><a href="#orgfbd9232">Equivalence class :</a></li> +</ul> +</li> +<li><a href="#org9a36dc1">Order relationship :</a> +<ul> +<li><a href="#orgb496cba"><span class="todo TODO">TODO</span> Examples :</a></li> +</ul> +</li> +</ul> +</li> +<li><a href="#org54d5489">TP exercices <i>Oct 20</i> :</a> +<ul> +<li><a href="#orgdfd55ca">Exercice 3 :</a> +<ul> +<li><a href="#org4100fe3">Question 3</a></li> +</ul> +</li> +<li><a href="#org019b5e0">Exercice 4 :</a> +<ul> +<li><a href="#org2ae1181"><span class="done DONE">DONE</span> Question 1 :</a></li> +</ul> +</li> +</ul> +</li> +</ul> +</div> +</div> +<div id="outline-container-orgdb6601b" class="outline-2"> +<h2 id="orgdb6601b">Contenu de la Matiére</h2> +<div class="outline-text-2" id="text-orgdb6601b"> </div> -<div id="outline-container-org2a89be2" class="outline-3"> -<h3 id="org2a89be2">Rappels et compléments (11H)</h3> -<div class="outline-text-3" id="text-org2a89be2"> +<div id="outline-container-orgc89165c" class="outline-3"> +<h3 id="orgc89165c">Rappels et compléments (11H)</h3> +<div class="outline-text-3" id="text-orgc89165c"> <ul class="org-ul"> -<li>Logique mathématique et méthodes du raisonnement mathématique</li> -<li>Ensembles et Relations</li> -<li>Applications</li> +<li>Logique mathématique et méthodes du raisonnement mathématique<br /></li> +<li>Ensembles et Relations<br /></li> +<li>Applications<br /></li> </ul> </div> </div> -<div id="outline-container-orgfcd7f3f" class="outline-3"> -<h3 id="orgfcd7f3f">Structures Algébriques (11H)</h3> -<div class="outline-text-3" id="text-orgfcd7f3f"> +<div id="outline-container-org1ae59da" class="outline-3"> +<h3 id="org1ae59da">Structures Algébriques (11H)</h3> +<div class="outline-text-3" id="text-org1ae59da"> <ul class="org-ul"> -<li>Groupes et morphisme de groupes</li> -<li>Anneaux et morphisme d’anneaux</li> -<li>Les corps</li> +<li>Groupes et morphisme de groupes<br /></li> +<li>Anneaux et morphisme d’anneaux<br /></li> +<li>Les corps<br /></li> </ul> </div> </div> -<div id="outline-container-org03cbf05" class="outline-3"> -<h3 id="org03cbf05">Polynômes et fractions rationnelles</h3> -<div class="outline-text-3" id="text-org03cbf05"> +<div id="outline-container-orgdcb51e0" class="outline-3"> +<h3 id="orgdcb51e0">Polynômes et fractions rationnelles</h3> +<div class="outline-text-3" id="text-orgdcb51e0"> <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> +<li>Notion du polynôme à une indéterminée á coefficients dans un anneau<br /></li> +<li>Opérations Algébriques sur les polynômes<br /></li> +<li>Arithmétique dans l’anneau des polynômes<br /></li> +<li>Polynôme dérivé et formule de Taylor<br /></li> +<li>Notion de racine d’un polynôme<br /></li> +<li>Notion de Fraction rationelle á une indéterminée<br /></li> +<li>Décomposition des fractions rationelles en éléments simples<br /></li> </ul> </div> </div> </div> -<div id="outline-container-org7db21e0" class="outline-2"> -<h2 id="org7db21e0">Premier cours : Logique mathématique et méthodes du raisonnement mathématique <i>Sep 25</i> :</h2> -<div class="outline-text-2" id="text-org7db21e0"> +<div id="outline-container-org8356dfb" class="outline-2"> +<h2 id="org8356dfb">Premier cours : Logique mathématique et méthodes du raisonnement mathématique <i>Sep 25</i> :</h2> +<div class="outline-text-2" id="text-org8356dfb"> <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. +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> <p> -<i>Ex:</i> +<i>Ex:</i><br /> </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> +<li><b>5 ≥ 2</b> is a proposition, a correct one !!!<br /></li> +<li><b>The webmaster is a girl</b> is also a proposition, which is also correct.<br /></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.<br /></li> </ul> <p> -…etc +…etc<br /> </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>. +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>.<br /> </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> +So now we could write :<br /> +<b>Let the proposition P be 5 ≥ 2, we notice that P is always True, therefor its validity is 1</b><br /> </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: +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:<br /> </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> +<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><br /> </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 +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<br /> </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> +Ex:<br /> +<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><br /> </p> <table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> @@ -176,23 +285,23 @@ Ex: </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> +<i>What the hell is this ?</i><br /> +The first colomn is equivalent to saying : “The webmaster is asleep AND The reader loves pufferfishes”<br /> +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>)”<br /> +The third one… <i>zzzzzzz</i><br /> </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. +You got the idea !!!<br /> +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.<br /> </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> +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><br /> </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 : +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 :<br /> </p> <table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> @@ -247,28 +356,28 @@ What if, we want to test whether or not two propositions are true AT THE SAME TI </table> <p> -<b>Always remember: 1 means true and 0 means false</b> +<b>Always remember: 1 means true and 0 means false</b><br /> </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> +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><br /> </p> <p> -Implication is kinda hard for my little brain to explain, so I will just say what it means: +Implication is kinda hard for my little brain to explain, so I will just say what it means:<br /> </p> <p> -<b>If P implies Q, this means that either Q, or the opposite of P are correct</b> +<b>If P implies Q, this means that either Q, or the opposite of P are correct</b><br /> </p> <p> -or in math terms +or in math terms<br /> </p> <p> -<b>P ⇒ Q translates to P̅ ∨ Q</b> -Let’s illustrate : +<b>P ⇒ Q translates to P̅ ∨ Q</b><br /> +Let’s illustrate :<br /> </p> <table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> @@ -344,15 +453,15 @@ Let’s illustrate : </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> +<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><br /> </p> <p> -Aight, a last one and we are done!!! Equivalence, which is fairly easy, symbolized by a <b>⇔</b> symbol. +Aight, a last one and we are done!!! Equivalence, which is fairly easy, symbolized by a <b>⇔</b> symbol.<br /> </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. +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.<br /> </p> <table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides"> @@ -435,332 +544,332 @@ A proposition is equivalent to another only when both of them have <b>the same v </table> <p> -<i>Note: P implying Q is equivalent to P̅ implying Q̅, or: (P ⇒ Q) ⇔ (P̅ ⇒ Q̅)</i> +<i>Note: P implying Q is equivalent to P̅ implying Q̅, or: (P ⇒ Q) ⇔ (P̅ ⇒ Q̅)</i><br /> </p> </div> -<div id="outline-container-org5604636" class="outline-3"> -<h3 id="org5604636">Properties:</h3> -<div class="outline-text-3" id="text-org5604636"> +<div id="outline-container-org4521340" class="outline-3"> +<h3 id="org4521340">Properties:</h3> +<div class="outline-text-3" id="text-org4521340"> </div> -<div id="outline-container-orgfffc23d" class="outline-4"> -<h4 id="orgfffc23d"><b>Absorption</b>:</h4> -<div class="outline-text-4" id="text-orgfffc23d"> +<div id="outline-container-org84e1ec3" class="outline-4"> +<h4 id="org84e1ec3"><b>Absorption</b>:</h4> +<div class="outline-text-4" id="text-org84e1ec3"> <p> -(P ∨ P) ⇔ P +(P ∨ P) ⇔ P<br /> </p> <p> -(P ∧ P) ⇔ P +(P ∧ P) ⇔ P<br /> </p> </div> </div> -<div id="outline-container-orgd43aeb7" class="outline-4"> -<h4 id="orgd43aeb7"><b>Commutativity</b>:</h4> -<div class="outline-text-4" id="text-orgd43aeb7"> +<div id="outline-container-org2345992" class="outline-4"> +<h4 id="org2345992"><b>Commutativity</b>:</h4> +<div class="outline-text-4" id="text-org2345992"> <p> -(P ∧ Q) ⇔ (Q ∧ P) +(P ∧ Q) ⇔ (Q ∧ P)<br /> </p> <p> -(P ∨ Q) ⇔ (Q ∨ P) +(P ∨ Q) ⇔ (Q ∨ P)<br /> </p> </div> </div> -<div id="outline-container-org9e5868e" class="outline-4"> -<h4 id="org9e5868e"><b>Associativity</b>:</h4> -<div class="outline-text-4" id="text-org9e5868e"> +<div id="outline-container-orgfee557e" class="outline-4"> +<h4 id="orgfee557e"><b>Associativity</b>:</h4> +<div class="outline-text-4" id="text-orgfee557e"> <p> -P ∧ (Q ∧ R) ⇔ (P ∧ Q) ∧ R +P ∧ (Q ∧ R) ⇔ (P ∧ Q) ∧ R<br /> </p> <p> -P ∨ (Q ∨ R) ⇔ (P ∨ Q) ∨ R +P ∨ (Q ∨ R) ⇔ (P ∨ Q) ∨ R<br /> </p> </div> </div> -<div id="outline-container-orga530d13" class="outline-4"> -<h4 id="orga530d13"><b>Distributivity</b>:</h4> -<div class="outline-text-4" id="text-orga530d13"> +<div id="outline-container-orgb55cfb2" class="outline-4"> +<h4 id="orgb55cfb2"><b>Distributivity</b>:</h4> +<div class="outline-text-4" id="text-orgb55cfb2"> <p> -P ∧ (Q ∨ R) ⇔ (P ∧ Q) ∨ (P ∧ R) +P ∧ (Q ∨ R) ⇔ (P ∧ Q) ∨ (P ∧ R)<br /> </p> <p> -P ∨ (Q ∧ R) ⇔ (P ∨ Q) ∧ (P ∨ R) +P ∨ (Q ∧ R) ⇔ (P ∨ Q) ∧ (P ∨ R)<br /> </p> </div> </div> -<div id="outline-container-org7d55048" class="outline-4"> -<h4 id="org7d55048"><b>Neutral element</b>:</h4> -<div class="outline-text-4" id="text-org7d55048"> +<div id="outline-container-orgb0de2bb" class="outline-4"> +<h4 id="orgb0de2bb"><b>Neutral element</b>:</h4> +<div class="outline-text-4" id="text-orgb0de2bb"> <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> +<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> <p> -P ∧ T ⇔ P +P ∧ T ⇔ P<br /> </p> <p> -P ∨ F ⇔ P +P ∨ F ⇔ P<br /> </p> </div> </div> -<div id="outline-container-org7422610" class="outline-4"> -<h4 id="org7422610"><b>Negation of a conjunction & a disjunction</b>:</h4> -<div class="outline-text-4" id="text-org7422610"> +<div id="outline-container-orgb51e625" class="outline-4"> +<h4 id="orgb51e625"><b>Negation of a conjunction & a disjunction</b>:</h4> +<div class="outline-text-4" id="text-orgb51e625"> <p> -Now we won’t use bars here because my lazy ass doesn’t know how, so instead I will use not()!!! +Now we won’t use bars here because my lazy ass doesn’t know how, so instead I will use not()!!!<br /> </p> <p> -not(<b>P ∧ Q</b>) ⇔ P̅ ∨ Q̅ +not(<b>P ∧ Q</b>) ⇔ P̅ ∨ Q̅<br /> </p> <p> -not(<b>P ∨ Q</b>) ⇔ P̅ ∧ Q̅ +not(<b>P ∨ Q</b>) ⇔ P̅ ∧ Q̅<br /> </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> +<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><br /> </p> </div> </div> -<div id="outline-container-org4145760" class="outline-4"> -<h4 id="org4145760"><b>Transitivity</b>:</h4> -<div class="outline-text-4" id="text-org4145760"> +<div id="outline-container-org4e29124" class="outline-4"> +<h4 id="org4e29124"><b>Transitivity</b>:</h4> +<div class="outline-text-4" id="text-org4e29124"> <p> -[(P ⇒ Q) AND (Q ⇒ R)] ⇔ P ⇒ R +[(P ⇒ Q) AND (Q ⇒ R)] ⇔ P ⇒ R<br /> </p> </div> </div> -<div id="outline-container-org245af1d" class="outline-4"> -<h4 id="org245af1d"><b>Contraposition</b>:</h4> -<div class="outline-text-4" id="text-org245af1d"> +<div id="outline-container-org520b7b0" class="outline-4"> +<h4 id="org520b7b0"><b>Contraposition</b>:</h4> +<div class="outline-text-4" id="text-org520b7b0"> <p> -(P ⇒ Q) ⇔ (Q̅ ⇒ P̅) +(P ⇒ Q) ⇔ (Q̅ ⇒ P̅)<br /> </p> </div> </div> -<div id="outline-container-orga47b617" class="outline-4"> -<h4 id="orga47b617">God only knows what this property is called:</h4> -<div class="outline-text-4" id="text-orga47b617"> +<div id="outline-container-org8159635" class="outline-4"> +<h4 id="org8159635">God only knows what this property is called:</h4> +<div class="outline-text-4" id="text-org8159635"> <p> -<i>If</i> +<i>If</i><br /> </p> <p> -(P ⇒ Q) is true +(P ⇒ Q) is true<br /> </p> <p> -and +and<br /> </p> <p> -(P̅ ⇒ Q) is true +(P̅ ⇒ Q) is true<br /> </p> <p> -then +then<br /> </p> <p> -Q is always true +Q is always true<br /> </p> </div> </div> </div> -<div id="outline-container-org3cfbd88" class="outline-3"> -<h3 id="org3cfbd88">Some exercices I found online :</h3> -<div class="outline-text-3" id="text-org3cfbd88"> +<div id="outline-container-org4210e18" class="outline-3"> +<h3 id="org4210e18">Some exercices I found online :</h3> +<div class="outline-text-3" id="text-org4210e18"> </div> -<div id="outline-container-orge60008b" class="outline-4"> -<h4 id="orge60008b">USTHB 2022/2023 Section B :</h4> -<div class="outline-text-4" id="text-orge60008b"> +<div id="outline-container-org46d1ee6" class="outline-4"> +<h4 id="org46d1ee6">USTHB 2022/2023 Section B :</h4> +<div class="outline-text-4" id="text-org46d1ee6"> </div> <ul class="org-ul"> -<li><a id="orgd7d6ce9"></a>Exercice 1: Démontrer les équivalences suivantes:<br /> -<div class="outline-text-5" id="text-orgd7d6ce9"> +<li><a id="orga6d3248"></a>Exercice 1: Démontrer les équivalences suivantes:<br /> +<div class="outline-text-5" id="text-orga6d3248"> <ol class="org-ol"> <li><p> -(P ⇒ Q) ⇔ (Q̅ ⇒ P̅) +(P ⇒ Q) ⇔ (Q̅ ⇒ P̅)<br /> </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> +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><br /> </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 : +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 :<br /> </p> <p> -<b>(Q ∨ P̅) ⇔ (P̅ ∨ Q)</b>, which is true because of commutativity +<b>(Q ∨ P̅) ⇔ (P̅ ∨ Q)</b>, which is true because of commutativity<br /> </p> <ol class="org-ol"> -<li>not(P ⇒ Q) ⇔ P ∧ Q̅</li> +<li>not(P ⇒ Q) ⇔ P ∧ Q̅<br /></li> </ol> <p> -Okaaaay so, let’s first get rid of the implication, because I don’t like it : <b>not(P̅ ∨ Q)</b> +Okaaaay so, let’s first get rid of the implication, because I don’t like it : <b>not(P̅ ∨ Q)</b><br /> </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 +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<br /> </p> <ol class="org-ol"> <li><p> -P ⇒ (Q ∧ R) ⇔ (P ⇒ Q) ∧ (P ⇒ R) +P ⇒ (Q ∧ R) ⇔ (P ⇒ Q) ∧ (P ⇒ R)<br /> </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 +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<br /> </p></li> <li><p> -P ∧ (Q ∨ R) ⇔ (P ∧ Q) ∨ (P ∧ R) +P ∧ (Q ∨ R) ⇔ (P ∧ Q) ∨ (P ∧ R)<br /> </p> <p> -Literally the same as above 🩷 +Literally the same as above 🩷<br /> </p></li> </ol> </div> </li> -<li><a id="orgd64e49a"></a>Exercice 2: Dire si les propositions suivantes sont vraies ou fausses, et les nier:<br /> -<div class="outline-text-5" id="text-orgd64e49a"> +<li><a id="orgb4b0c43"></a>Exercice 2: Dire si les propositions suivantes sont vraies ou fausses, et les nier:<br /> +<div class="outline-text-5" id="text-orgb4b0c43"> <ol class="org-ol"> <li><p> -∀x ∈ ℝ ,∃y ∈ ℝ*+, tels que e^x = y +∀x ∈ ℝ ,∃y ∈ ℝ*+, tels que e^x = y<br /> </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 +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<br /> </p></li> </ol> <p> -“The function f(x)=e^x is always positive and non-null”, the very definition of an exponential function !!!! +“The function f(x)=e^x is always positive and non-null”, the very definition of an exponential function !!!!<br /> </p> <p> -<b>So the proposition is true</b> +<b>So the proposition is true</b><br /> </p> <ol class="org-ol"> -<li>∃x ∈ ℝ, tels que x^2 < x < x^3</li> +<li>∃x ∈ ℝ, tels que x^2 < x < x^3<br /></li> </ol> <p> -We just need to find a value that satisifies this condition…thankfully its easy…. +We just need to find a value that satisifies this condition…thankfully its easy….<br /> </p> <p> -x² < x < x³ , we divide the three terms by x so we get : +x² < x < x³ , we divide the three terms by x so we get :<br /> </p> <p> -x < 1 < x² , or : +x < 1 < x² , or :<br /> </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> +<b>x < 1</b> ; <b>1 < x²</b> ⇔ <b>x < 1</b> ; <b>1 < x</b> <i>We square root both sides</i><br /> </p> <p> -We end up with a contradiction, therefor its wrong +We end up with a contradiction, therefor its wrong<br /> </p> <ol class="org-ol"> -<li>∀x ∈ ℝ, ∃y ∈ ℝ tels que y = 3x - 8</li> +<li>∀x ∈ ℝ, ∃y ∈ ℝ tels que y = 3x - 8<br /></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… +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…<br /> </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> +<b>Since substraction is an Internal composition law in ℝ, therefor all results of a substraction between two Real numbers is…Real</b><br /> </p> <ol class="org-ol"> <li><p> -∃x ∈ ℕ, ∀y ∈ ℕ, x > y ⇒ x + y < 8 +∃x ∈ ℕ, ∀y ∈ ℕ, x > y ⇒ x + y < 8<br /> </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” +“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”<br /> </p></li> </ol> <p> -Let’s get rid of the implication : +Let’s get rid of the implication :<br /> </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> +∃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><br /> </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: +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:<br /> </p> <p> -<b>x = 3 , if y > 3 then the first condition is true ; if y < 3 then the second one is true</b> +<b>x = 3 , if y > 3 then the first condition is true ; if y < 3 then the second one is true</b><br /> </p> <p> -Meaning that the two propositions CAN NOT BE WRONG TOGETHER, either one is wrong, or the other +Meaning that the two propositions CAN NOT BE WRONG TOGETHER, either one is wrong, or the other<br /> </p> <p> -y > x +y > x<br /> </p> <p> -<b>y - x > 0</b> +<b>y - x > 0</b><br /> </p> <p> -y + x < 8 +y + x < 8<br /> </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> +<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><br /> </p> <ol class="org-ol"> <li><p> -∀x ∈ ℝ, x² ≥ 1 ⇔ x ≥ 1 +∀x ∈ ℝ, x² ≥ 1 ⇔ x ≥ 1<br /> </p> <p> -….This is getting stupid. of course it’s true it’s part of the definition of the power of 2 +….This is getting stupid. of course it’s true it’s part of the definition of the power of 2<br /> </p></li> </ol> </div> @@ -769,684 +878,745 @@ y + x < 8 </div> </div> </div> -<div id="outline-container-org980d3be" class="outline-2"> -<h2 id="org980d3be">2éme cours <i>Oct 2</i></h2> -<div class="outline-text-2" id="text-org980d3be"> +<div id="outline-container-org732b9dd" class="outline-2"> +<h2 id="org732b9dd">2éme cours <i>Oct 2</i></h2> +<div class="outline-text-2" id="text-org732b9dd"> </div> -<div id="outline-container-org22b148b" class="outline-3"> -<h3 id="org22b148b">Quantifiers</h3> -<div class="outline-text-3" id="text-org22b148b"> +<div id="outline-container-orgdfba00a" class="outline-3"> +<h3 id="orgdfba00a">Quantifiers</h3> +<div class="outline-text-3" id="text-orgdfba00a"> <p> -A propriety P can depend on a parameter x +A propriety P can depend on a parameter x<br /> </p> <p> -∀ is the universal quantifier which stands for “For any value of…” +∀ is the universal quantifier which stands for “For any value of…”<br /> </p> <p> -∃ is the existential quantifier which stands for “There exists at least one…” +∃ is the existential quantifier which stands for “There exists at least one…”<br /> </p> </div> <ul class="org-ul"> -<li><a id="org4afa1df"></a>Example<br /> -<div class="outline-text-6" id="text-org4afa1df"> +<li><a id="org9a107f8"></a>Example<br /> +<div class="outline-text-6" id="text-org9a107f8"> <p> -P(x) : x+1≥0 +P(x) : x+1≥0<br /> </p> <p> -P(X) is True or False depending on the values of x +P(X) is True or False depending on the values of x<br /> </p> </div> </li> </ul> -<div id="outline-container-org8b437f3" class="outline-4"> -<h4 id="org8b437f3">Proprieties</h4> -<div class="outline-text-4" id="text-org8b437f3"> +<div id="outline-container-org93d5891" class="outline-4"> +<h4 id="org93d5891">Proprieties</h4> +<div class="outline-text-4" id="text-org93d5891"> </div> <ul class="org-ul"> -<li><a id="org6d0c06f"></a>Propriety Number 1:<br /> -<div class="outline-text-5" id="text-org6d0c06f"> +<li><a id="orgcc6c2bd"></a>Propriety Number 1:<br /> +<div class="outline-text-5" id="text-orgcc6c2bd"> <p> -The negation of the universal quantifier is the existential quantifier, and vice-versa : +The negation of the universal quantifier is the existential quantifier, and vice-versa :<br /> </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> +<li>not(∀x ∈ E , P(x)) ⇔ ∃ x ∈ E, not(P(x))<br /></li> +<li>not(∃x ∈ E , P(x)) ⇔ ∀ x ∈ E, not(P(x))<br /></li> </ul> </div> <ul class="org-ul"> -<li><a id="orgd242e81"></a>Example:<br /> -<div class="outline-text-6" id="text-orgd242e81"> +<li><a id="org4d384a3"></a>Example:<br /> +<div class="outline-text-6" id="text-org4d384a3"> <p> -∀ x ≥ 1 x² > 5 ⇔ ∃ x ≥ 1 x² < 5 +∀ x ≥ 1 x² > 5 ⇔ ∃ x ≥ 1 x² < 5<br /> </p> </div> </li> </ul> </li> -<li><a id="orgd7c2c1d"></a>Propriety Number 2:<br /> -<div class="outline-text-5" id="text-orgd7c2c1d"> +<li><a id="org372a28e"></a>Propriety Number 2:<br /> +<div class="outline-text-5" id="text-org372a28e"> <p> -<b>∀x ∈ E, [P(x) ∧ Q(x)] ⇔ [∀ x ∈ E, P(x)] ∧ [∀ x ∈ E, Q(x)]</b> +<b>∀x ∈ E, [P(x) ∧ Q(x)] ⇔ [∀ x ∈ E, P(x)] ∧ [∀ x ∈ E, Q(x)]</b><br /> </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)” +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)”<br /> </p> </div> <ul class="org-ul"> -<li><a id="org5cb6921"></a>Example :<br /> -<div class="outline-text-6" id="text-org5cb6921"> +<li><a id="org7649498"></a>Example :<br /> +<div class="outline-text-6" id="text-org7649498"> <p> -P(x) : sqrt(x) > 0 ; Q(x) : x ≥ 1 +P(x) : sqrt(x) > 0 ; Q(x) : x ≥ 1<br /> </p> <p> -∀x ∈ ℝ*+, [sqrt(x) > 0 , x ≥ 1] ⇔ [∀x ∈ R*+, sqrt(x) > 0] ∧ [∀x ∈ R*+, x ≥ 1] +∀x ∈ ℝ*+, [sqrt(x) > 0 , x ≥ 1] ⇔ [∀x ∈ R*+, sqrt(x) > 0] ∧ [∀x ∈ R*+, x ≥ 1]<br /> </p> <p> -<b>Which is true</b> +<b>Which is true</b><br /> </p> </div> </li> </ul> </li> -<li><a id="orgd56cb14"></a>Propriety Number 3:<br /> -<div class="outline-text-5" id="text-orgd56cb14"> +<li><a id="orgbccb33f"></a>Propriety Number 3:<br /> +<div class="outline-text-5" id="text-orgbccb33f"> <p> -<b>∃ x ∈ E, [P(x) ∧ Q(x)] <i>⇒</i> [∃ x ∈ E, P(x)] ∧ [∃ x ∈ E, Q(x)]</b> +<b>∃ x ∈ E, [P(x) ∧ Q(x)] <i>⇒</i> [∃ x ∈ E, P(x)] ∧ [∃ x ∈ E, Q(x)]</b><br /> </p> <p> -<i>Here its an implication and not an equivalence</i> +<i>Here its an implication and not an equivalence</i><br /> </p> </div> <ul class="org-ul"> -<li><a id="org52c3098"></a>Example of why it’s NOT an equivalence :<br /> -<div class="outline-text-6" id="text-org52c3098"> +<li><a id="orgf623821"></a>Example of why it’s NOT an equivalence :<br /> +<div class="outline-text-6" id="text-orgf623821"> <p> -P(x) : x > 5 ; Q(x) : x < 5 +P(x) : x > 5 ; Q(x) : x < 5<br /> </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!!! +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!!!<br /> </p> </div> </li> </ul> </li> -<li><a id="org9439534"></a>Propriety Number 4:<br /> -<div class="outline-text-5" id="text-org9439534"> +<li><a id="orgda152b2"></a>Propriety Number 4:<br /> +<div class="outline-text-5" id="text-orgda152b2"> <p> -<b>[∀ x ∈ E, P(x)] ∨ [∀ x ∈ E, Q(x)] <i>⇒</i> ∀x ∈ E, [P(x) ∨ Q(x)]</b> +<b>[∀ x ∈ E, P(x)] ∨ [∀ x ∈ E, Q(x)] <i>⇒</i> ∀x ∈ E, [P(x) ∨ Q(x)]</b><br /> </p> <p> -<i>Same here, implication and NOT en equivalence</i> +<i>Same here, implication and NOT en equivalence</i><br /> </p> </div> </li> </ul> </div> </div> -<div id="outline-container-orgcb2ff75" class="outline-3"> -<h3 id="orgcb2ff75">Multi-parameter proprieties :</h3> -<div class="outline-text-3" id="text-orgcb2ff75"> +<div id="outline-container-orgf21a239" class="outline-3"> +<h3 id="orgf21a239">Multi-parameter proprieties :</h3> +<div class="outline-text-3" id="text-orgf21a239"> <p> -A propriety P can depend on two or more parameters, for convenience we call them x,y,z…etc +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="org309a152"></a>Example :<br /> -<div class="outline-text-6" id="text-org309a152"> +<li><a id="orgdda5feb"></a>Example :<br /> +<div class="outline-text-6" id="text-orgdda5feb"> <p> -P(x,y): x+y > 0 +P(x,y): x+y > 0<br /> </p> <p> -P(0,1) is a True proposition +P(0,1) is a True proposition<br /> </p> <p> -P(-2,-1) is a False one +P(-2,-1) is a False one<br /> </p> </div> </li> -<li><a id="orgfbf5cee"></a>WARNING :<br /> -<div class="outline-text-6" id="text-orgfbf5cee"> +<li><a id="org08d728b"></a>WARNING :<br /> +<div class="outline-text-6" id="text-org08d728b"> <p> -∀x ∈ E, ∃y ∈ F , P(x,y) +∀x ∈ E, ∃y ∈ F , P(x,y)<br /> </p> <p> -∃y ∈ F, ∀x ∈ E , P(x,y) +∃y ∈ F, ∀x ∈ E , P(x,y)<br /> </p> <p> -Are different because in the first one y depends on x, while in the second one, it doesn’t +Are different because in the first one y depends on x, while in the second one, it doesn’t<br /> </p> </div> <ul class="org-ul"> -<li><a id="org21332e2"></a>Example :<br /> -<div class="outline-text-7" id="text-org21332e2"> +<li><a id="org78dcf22"></a>Example :<br /> +<div class="outline-text-7" id="text-org78dcf22"> <p> -∀ x ∈ ℕ , ∃ y ∈ ℕ y > x -–— True +∀ x ∈ ℕ , ∃ y ∈ ℕ y > x -–— True<br /> </p> <p> -∃ y ∈ ℕ , ∀ x ∈ ℕ y > x -–— False +∃ y ∈ ℕ , ∀ x ∈ ℕ y > x -–— False<br /> </p> </div> </li> </ul> </li> </ul> -<li><a id="org2fad1a6"></a>Proprieties :<br /> -<div class="outline-text-5" id="text-org2fad1a6"> +<li><a id="org088c862"></a>Proprieties :<br /> +<div class="outline-text-5" id="text-org088c862"> <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> +<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> </ol> </div> </li> </ul> </div> -<div id="outline-container-org405d91a" class="outline-3"> -<h3 id="org405d91a">Methods of mathematical reasoning :</h3> -<div class="outline-text-3" id="text-org405d91a"> +<div id="outline-container-org779f917" class="outline-3"> +<h3 id="org779f917">Methods of mathematical reasoning :</h3> +<div class="outline-text-3" id="text-org779f917"> </div> -<div id="outline-container-org0e2120a" class="outline-4"> -<h4 id="org0e2120a">Direct reasoning :</h4> -<div class="outline-text-4" id="text-org0e2120a"> +<div id="outline-container-orgc6832c3" class="outline-4"> +<h4 id="orgc6832c3">Direct reasoning :</h4> +<div class="outline-text-4" id="text-orgc6832c3"> <p> -To show that an implication P ⇒ Q is true, we suppose that P is true and we show that Q is true +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="orge655791"></a>Example:<br /> -<div class="outline-text-5" id="text-orge655791"> +<li><a id="org8dbaab8"></a>Example:<br /> +<div class="outline-text-5" id="text-org8dbaab8"> <p> -Let a,b be two Real numbers, we have to prove that <b>a² + b² = 1 ⇒ |a + b| ≤ 2</b> +Let a,b be two Real numbers, we have to prove that <b>a² + b² = 1 ⇒ |a + b| ≤ 2</b><br /> </p> <p> -We suppose that a²+b² = 1 and we prove that |a + b| ≤ 2 +We suppose that a²+b² = 1 and we prove that |a + b| ≤ 2<br /> </p> <p> -a²+b²=1 ⇒ b² = 1 - a² ; a² = 1 - b² +a²+b²=1 ⇒ b² = 1 - a² ; a² = 1 - b²<br /> </p> <p> -a²+b²=1 ⇒ 1 - a² ≥ 0 ; 1 - b² ≥ 0 +a²+b²=1 ⇒ 1 - a² ≥ 0 ; 1 - b² ≥ 0<br /> </p> <p> -a²+b²=1 ⇒ a² ≤ 1 ; b² ≤ 1 +a²+b²=1 ⇒ a² ≤ 1 ; b² ≤ 1<br /> </p> <p> -a²+b²=1 ⇒ -1 ≤ a ≤ 1 ; -1 ≤ b ≤ 1 +a²+b²=1 ⇒ -1 ≤ a ≤ 1 ; -1 ≤ b ≤ 1<br /> </p> <p> -a²+b²=1 ⇒ -2 ≤ a + b ≤ 2 +a²+b²=1 ⇒ -2 ≤ a + b ≤ 2<br /> </p> <p> -a²+b²=1 ⇒ |a + b| ≤ 2 <b>Which is what we wanted to prove, therefor the implication is correct</b> +a²+b²=1 ⇒ |a + b| ≤ 2 <b>Which is what we wanted to prove, therefor the implication is correct</b><br /> </p> </div> </li> </ul> </div> -<div id="outline-container-org3318c18" class="outline-4"> -<h4 id="org3318c18">Reasoning by the Absurd:</h4> -<div class="outline-text-4" id="text-org3318c18"> +<div id="outline-container-orgf140b16" class="outline-4"> +<h4 id="orgf140b16">Reasoning by the Absurd:</h4> +<div class="outline-text-4" id="text-orgf140b16"> <p> -To prove that a proposition is True, we suppose that it’s False and we must come to a contradiction +To prove that a proposition is True, we suppose that it’s False and we must come to a contradiction<br /> </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 +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<br /> </p> </div> <ul class="org-ul"> -<li><a id="org6217ba8"></a>Example:<br /> -<div class="outline-text-5" id="text-org6217ba8"> +<li><a id="org1fefbfb"></a>Example:<br /> +<div class="outline-text-5" id="text-org1fefbfb"> <p> -Prove that this proposition is correct using the reasoning by the absurd : ∀x ∈ ℝ* , sqrt(1+x²) ≠ 1 + x²/2 +Prove that this proposition is correct using the reasoning by the absurd : ∀x ∈ ℝ* , sqrt(1+x²) ≠ 1 + x²/2<br /> </p> <p> -We assume that ∃ x ℝ* , sqrt(1+x²) = 1 + x²/2 +We assume that ∃ x ℝ* , sqrt(1+x²) = 1 + x²/2<br /> </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 +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<br /> </p> </div> </li> </ul> </div> -<div id="outline-container-orgdca4b33" class="outline-4"> -<h4 id="orgdca4b33">Reasoning by contraposition:</h4> -<div class="outline-text-4" id="text-orgdca4b33"> +<div id="outline-container-org320dc57" class="outline-4"> +<h4 id="org320dc57">Reasoning by contraposition:</h4> +<div class="outline-text-4" id="text-org320dc57"> <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 +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-org45373bc" class="outline-4"> -<h4 id="org45373bc">Reasoning by counter example:</h4> -<div class="outline-text-4" id="text-org45373bc"> +<div id="outline-container-org27943b2" class="outline-4"> +<h4 id="org27943b2">Reasoning by counter example:</h4> +<div class="outline-text-4" id="text-org27943b2"> <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 +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> </div> </div> </div> </div> -<div id="outline-container-org1ab01d8" class="outline-2"> -<h2 id="org1ab01d8">3eme Cours : <i>Oct 9</i></h2> -<div class="outline-text-2" id="text-org1ab01d8"> +<div id="outline-container-orgfddd579" class="outline-2"> +<h2 id="orgfddd579">3eme Cours : <i>Oct 9</i></h2> +<div class="outline-text-2" id="text-orgfddd579"> </div> -<div id="outline-container-orgc3cdd55" class="outline-4"> -<h4 id="orgc3cdd55">Reasoning by recurrence :</h4> -<div class="outline-text-4" id="text-orgc3cdd55"> +<div id="outline-container-org21eab57" class="outline-4"> +<h4 id="org21eab57">Reasoning by recurrence :</h4> +<div class="outline-text-4" id="text-org21eab57"> <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 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="org74698a3"></a>Example:<br /> -<div class="outline-text-5" id="text-org74698a3"> +<li><a id="org2a3aa46"></a>Example:<br /> +<div class="outline-text-5" id="text-org2a3aa46"> <p> -Let’s prove that ∀ n ≥ 1 , (n,k=1)Σk = [n(n+1)]/2 +Let’s prove that ∀ n ≥ 1 , (n,k=1)Σk = [n(n+1)]/2<br /> </p> <p> -P(n) : (n,k=1)Σk = [n(n+1)]/2 +P(n) : (n,k=1)Σk = [n(n+1)]/2<br /> </p> <p> -<b>Pour n = 1:</b> (1,k=1)Σk = 1 ; [n(n+1)]/2 = 1 . <b>So P(1) is true</b> +<b>Pour n = 1:</b> (1,k=1)Σk = 1 ; [n(n+1)]/2 = 1 . <b>So P(1) is true</b><br /> </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> +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><br /> </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> +(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><br /> </p> <p> -<b>Conclusion: ∀n ≥ 1 , (n,k=1)Σk = n(n+1)/2</b> +<b>Conclusion: ∀n ≥ 1 , (n,k=1)Σk = n(n+1)/2</b><br /> </p> </div> </li> </ul> </div> </div> -<div id="outline-container-org62bfe2a" class="outline-2"> -<h2 id="org62bfe2a">4eme Cours : Chapitre 2 : Sets and Operations</h2> -<div class="outline-text-2" id="text-org62bfe2a"> +<div id="outline-container-org155c9a9" class="outline-2"> +<h2 id="org155c9a9">4eme Cours : Chapitre 2 : Sets and Operations</h2> +<div class="outline-text-2" id="text-org155c9a9"> </div> -<div id="outline-container-org5c29bea" class="outline-3"> -<h3 id="org5c29bea">Definition of a set :</h3> -<div class="outline-text-3" id="text-org5c29bea"> +<div id="outline-container-org5bb0b39" class="outline-3"> +<h3 id="org5bb0b39">Definition of a set :</h3> +<div class="outline-text-3" id="text-org5bb0b39"> <p> -A set is a collection of objects that share the sane propriety +A set is a collection of objects that share the sane propriety<br /> </p> </div> </div> -<div id="outline-container-org7f4934f" class="outline-3"> -<h3 id="org7f4934f">Belonging, inclusion, and equality :</h3> -<div class="outline-text-3" id="text-org7f4934f"> +<div id="outline-container-org469078a" class="outline-3"> +<h3 id="org469078a">Belonging, inclusion, and equality :</h3> +<div class="outline-text-3" id="text-org469078a"> <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> +<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> +<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><br /></li> +<li>The empty set (symbolized by ∅) is a set without elements, and is included in all sets (by convention) : <b>∅ ⊂ E</b><br /></li> </ol> </div> </div> -<div id="outline-container-orgd439312" class="outline-3"> -<h3 id="orgd439312">Intersections and reunions :</h3> -<div class="outline-text-3" id="text-orgd439312"> +<div id="outline-container-org1461e60" class="outline-3"> +<h3 id="org1461e60">Intersections and reunions :</h3> +<div class="outline-text-3" id="text-org1461e60"> </div> -<div id="outline-container-org2eaf0a6" class="outline-4"> -<h4 id="org2eaf0a6">Intersection:</h4> -<div class="outline-text-4" id="text-org2eaf0a6"> +<div id="outline-container-orgd9db499" class="outline-4"> +<h4 id="orgd9db499">Intersection:</h4> +<div class="outline-text-4" id="text-orgd9db499"> <p> -E ∩ F = {x / x ∈ E AND x ∈ F} ; x ∈ E ∩ F ⇔ x ∈ F AND x ∈ F +E ∩ F = {x / x ∈ E AND x ∈ F} ; x ∈ E ∩ F ⇔ x ∈ F AND x ∈ F<br /> </p> <p> -x ∉ E ∩ F ⇔ x ∉ E OR x ∉ F +x ∉ E ∩ F ⇔ x ∉ E OR x ∉ F<br /> </p> </div> </div> -<div id="outline-container-org8bfbedf" class="outline-4"> -<h4 id="org8bfbedf">Union:</h4> -<div class="outline-text-4" id="text-org8bfbedf"> +<div id="outline-container-org0946751" class="outline-4"> +<h4 id="org0946751">Union:</h4> +<div class="outline-text-4" id="text-org0946751"> <p> -E ∪ F = {x / x ∈ E OR x ∈ F} ; x ∈ E ∪ F ⇔ x ∈ F OR x ∈ F +E ∪ F = {x / x ∈ E OR x ∈ F} ; x ∈ E ∪ F ⇔ x ∈ F OR x ∈ F<br /> </p> <p> -x ∉ E ∪ F ⇔ x ∉ E AND x ∉ F +x ∉ E ∪ F ⇔ x ∉ E AND x ∉ F<br /> </p> </div> </div> -<div id="outline-container-orgf5d7c25" class="outline-4"> -<h4 id="orgf5d7c25">Difference between two sets:</h4> -<div class="outline-text-4" id="text-orgf5d7c25"> +<div id="outline-container-org8fb5e39" class="outline-4"> +<h4 id="org8fb5e39">Difference between two sets:</h4> +<div class="outline-text-4" id="text-org8fb5e39"> <p> -E\F(Which is also written as : E - F) = {x / x ∈ E and x ∉ F} +E(Which is also written as : E - F) = {x / x ∈ E and x ∉ F}<br /> </p> </div> </div> -<div id="outline-container-org16f26ee" class="outline-4"> -<h4 id="org16f26ee">Complimentary set:</h4> -<div class="outline-text-4" id="text-org16f26ee"> +<div id="outline-container-org17ec98b" class="outline-4"> +<h4 id="org17ec98b">Complimentary set:</h4> +<div class="outline-text-4" id="text-org17ec98b"> <p> -If F ⊂ E. E - F is the complimentary of F in E. +If F ⊂ E. E - F is the complimentary of F in E.<br /> </p> <p> -FCE = {x /x ∈ E AND x ∉ F} <b>ONLY WHEN F IS A SUBSET OF E</b> +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-org67da9c0" class="outline-4"> -<h4 id="org67da9c0">Symentrical difference</h4> -<div class="outline-text-4" id="text-org67da9c0"> +<div id="outline-container-org61d7d25" class="outline-4"> +<h4 id="org61d7d25">Symentrical difference</h4> +<div class="outline-text-4" id="text-org61d7d25"> <p> -E Δ F = (E - F) ∪ (F - E) ; = (E ∪ F) - (E ∩ F) +E Δ F = (E - F) ∪ (F - E) ; = (E ∪ F) - (E ∩ F)<br /> </p> </div> </div> </div> -<div id="outline-container-org10858f6" class="outline-3"> -<h3 id="org10858f6">Proprieties :</h3> -<div class="outline-text-3" id="text-org10858f6"> +<div id="outline-container-org1e0c21a" class="outline-3"> +<h3 id="org1e0c21a">Proprieties :</h3> +<div class="outline-text-3" id="text-org1e0c21a"> <p> -Let E,F and G be 3 sets. We have : +Let E,F and G be 3 sets. We have :<br /> </p> </div> -<div id="outline-container-orgdeeff37" class="outline-4"> -<h4 id="orgdeeff37">Commutativity:</h4> -<div class="outline-text-4" id="text-orgdeeff37"> +<div id="outline-container-orgaed74b7" class="outline-4"> +<h4 id="orgaed74b7">Commutativity:</h4> +<div class="outline-text-4" id="text-orgaed74b7"> <p> -E ∩ F = F ∩ E -E ∪ F = F ∪ E +E ∩ F = F ∩ E<br /> +E ∪ F = F ∪ E<br /> </p> </div> </div> -<div id="outline-container-org6228f00" class="outline-4"> -<h4 id="org6228f00">Associativity:</h4> -<div class="outline-text-4" id="text-org6228f00"> +<div id="outline-container-org1d74822" class="outline-4"> +<h4 id="org1d74822">Associativity:</h4> +<div class="outline-text-4" id="text-org1d74822"> <p> -E ∩ (F ∩ G) = (E ∩ F) ∩ G -E ∪ (F ∪ G) = (E ∪ F) ∪ G +E ∩ (F ∩ G) = (E ∩ F) ∩ G<br /> +E ∪ (F ∪ G) = (E ∪ F) ∪ G<br /> </p> </div> </div> -<div id="outline-container-org2523e0e" class="outline-4"> -<h4 id="org2523e0e">Distributivity:</h4> -<div class="outline-text-4" id="text-org2523e0e"> +<div id="outline-container-org667f4dd" class="outline-4"> +<h4 id="org667f4dd">Distributivity:</h4> +<div class="outline-text-4" id="text-org667f4dd"> <p> -E ∩ (F ∪ G) = (E ∩ F) ∪ (E ∩ G) -E ∪ (F ∩ G) = (E ∪ F) ∩ (E ∪ G) +E ∩ (F ∪ G) = (E ∩ F) ∪ (E ∩ G)<br /> +E ∪ (F ∩ G) = (E ∪ F) ∩ (E ∪ G)<br /> </p> </div> </div> -<div id="outline-container-orgeb0c0a3" class="outline-4"> -<h4 id="orgeb0c0a3">Lois de Morgan:</h4> -<div class="outline-text-4" id="text-orgeb0c0a3"> +<div id="outline-container-org31c4f57" class="outline-4"> +<h4 id="org31c4f57">Lois de Morgan:</h4> +<div class="outline-text-4" id="text-org31c4f57"> <p> -If E ⊂ G and F ⊂ G ; +If E ⊂ G and F ⊂ G ;<br /> </p> <p> -(E ∩ F)CG = ECG ∪ FCG ; (E ∪ F)CG = ECG ∩ FCG +(E ∩ F)CG = ECG ∪ FCG ; (E ∪ F)CG = ECG ∩ FCG<br /> </p> </div> </div> -<div id="outline-container-orge638501" class="outline-4"> -<h4 id="orge638501">An other one:</h4> -<div class="outline-text-4" id="text-orge638501"> +<div id="outline-container-orgc30ba1d" class="outline-4"> +<h4 id="orgc30ba1d">An other one:</h4> +<div class="outline-text-4" id="text-orgc30ba1d"> <p> -E - (F ∩ G) = (E-F) ∪ (E-G) ; E - (F ∪ G) = (E-F) ∩ (E-G) +E - (F ∩ G) = (E-F) ∪ (E-G) ; E - (F ∪ G) = (E-F) ∩ (E-G)<br /> </p> </div> </div> -<div id="outline-container-orgfe0b562" class="outline-4"> -<h4 id="orgfe0b562">An other one:</h4> -<div class="outline-text-4" id="text-orgfe0b562"> +<div id="outline-container-orgac52c86" class="outline-4"> +<h4 id="orgac52c86">An other one:</h4> +<div class="outline-text-4" id="text-orgac52c86"> <p> -E ∩ ∅ = ∅ ; E ∪ ∅ = E +E ∩ ∅ = ∅ ; E ∪ ∅ = E<br /> </p> </div> </div> -<div id="outline-container-org48afea2" class="outline-4"> -<h4 id="org48afea2">And an other one:</h4> -<div class="outline-text-4" id="text-org48afea2"> +<div id="outline-container-orge0de23e" class="outline-4"> +<h4 id="orge0de23e">And an other one:</h4> +<div class="outline-text-4" id="text-orge0de23e"> <p> -E ∩ (F Δ G) = (E ∩ F) Δ (E ∩ G) +E ∩ (F Δ G) = (E ∩ F) Δ (E ∩ G)<br /> </p> </div> </div> -<div id="outline-container-org1138be8" class="outline-4"> -<h4 id="org1138be8">And the last one:</h4> -<div class="outline-text-4" id="text-org1138be8"> +<div id="outline-container-org8277b0b" class="outline-4"> +<h4 id="org8277b0b">And the last one:</h4> +<div class="outline-text-4" id="text-org8277b0b"> <p> -E Δ ∅ = E ; E Δ E = ∅ +E Δ ∅ = E ; E Δ E = ∅<br /> </p> </div> </div> </div> </div> -<div id="outline-container-orgf188863" class="outline-2"> -<h2 id="orgf188863">5eme cours: L’ensemble des parties d’un ensemble <i>Oct 16</i></h2> -<div class="outline-text-2" id="text-orgf188863"> +<div id="outline-container-orgeb2675e" class="outline-2"> +<h2 id="orgeb2675e">5eme cours: L’ensemble des parties d’un ensemble <i>Oct 16</i></h2> +<div class="outline-text-2" id="text-orgeb2675e"> <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> +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-org6cfe0d7" class="outline-4"> -<h4 id="org6cfe0d7">Notes :</h4> -<div class="outline-text-4" id="text-org6cfe0d7"> +<div id="outline-container-org2c7d514" class="outline-4"> +<h4 id="org2c7d514">Notes :</h4> +<div class="outline-text-4" id="text-org2c7d514"> <p> -∅ ∈ P(E) ; E ∈ P(E) +∅ ∈ P(E) ; E ∈ P(E)<br /> </p> <p> -cardinal E = n <i>The number of terms in E</i> , cardinal P(E) = 2^n <i>The number of all parts of E</i> +cardinal E = n <i>The number of terms in E</i> , cardinal P(E) = 2^n <i>The number of all parts of E</i><br /> </p> </div> </div> -<div id="outline-container-orgd0b341d" class="outline-4"> -<h4 id="orgd0b341d">Examples :</h4> -<div class="outline-text-4" id="text-orgd0b341d"> +<div id="outline-container-org76da8f4" class="outline-4"> +<h4 id="org76da8f4">Examples :</h4> +<div class="outline-text-4" id="text-org76da8f4"> <p> -E = {a,b,c} // P(E)={∅, {a}, {b}, {c}, {a,b}, {b,c}, {a,c}, {a,b,c}} +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-org7ec7b74" class="outline-3"> -<h3 id="org7ec7b74">Partition of a set :</h3> -<div class="outline-text-3" id="text-org7ec7b74"> +<div id="outline-container-org3da3de1" class="outline-3"> +<h3 id="org3da3de1">Partition of a set :</h3> +<div class="outline-text-3" id="text-org3da3de1"> <p> -We say that <b>A</b> is a partition of E if: +We say that <b>A</b> is a partition of E if:<br /> </p> <ol class="org-ol"> -<li>∀ x ∈ A , x ≠ 0</li> -<li>All the elements of <b>A</b> are two by two disjoint. Or in other terms, there should not be two elements that intersects with each other.</li> -<li>The reunion of all elements of <b>A</b> is equal to E</li> +<li>∀ x ∈ A , x ≠ 0<br /></li> +<li>All the elements of <b>A</b> are two by two disjoint. Or in other terms, there should not be two elements that intersects with each other.<br /></li> +<li>The reunion of all elements of <b>A</b> is equal to E<br /></li> </ol> </div> </div> -<div id="outline-container-orgc0fd081" class="outline-3"> -<h3 id="orgc0fd081">Cartesian products :</h3> -<div class="outline-text-3" id="text-orgc0fd081"> +<div id="outline-container-org077b994" class="outline-3"> +<h3 id="org077b994">Cartesian products :</h3> +<div class="outline-text-3" id="text-org077b994"> <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 +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-org4b0f328" class="outline-4"> -<h4 id="org4b0f328">Example :</h4> -<div class="outline-text-4" id="text-org4b0f328"> +<div id="outline-container-org72822f5" class="outline-4"> +<h4 id="org72822f5">Example :</h4> +<div class="outline-text-4" id="text-org72822f5"> <p> -A = {4,5} ; B= {4,5,6} // AxB = {(4,4), (4,5), (4,6), (5,4), (5,5), (5,6)} +A = {4,5} ; B= {4,5,6} ; AxB = {(4,4), (4,5), (4,6), (5,4), (5,5), (5,6)}<br /> </p> <p> -BxA = {(4,4), (4,5), (5,4), (5,5), (6,4), (6,5)} // Therefore AxB ≠ BxA +BxA = {(4,4), (4,5), (5,4), (5,5), (6,4), (6,5)} ; Therefore AxB ≠ BxA<br /> </p> </div> </div> -<div id="outline-container-orgc924520" class="outline-4"> -<h4 id="orgc924520">Some proprieties:</h4> -<div class="outline-text-4" id="text-orgc924520"> +<div id="outline-container-org04e1be3" class="outline-4"> +<h4 id="org04e1be3">Some proprieties:</h4> +<div class="outline-text-4" id="text-org04e1be3"> <ol class="org-ol"> -<li>ExF = ∅ ⇔ E=∅ OR F=∅</li> -<li>ExF = FxE ⇔ E=F OR E=∅ OR F=∅</li> -<li>E x (F∪G) = (ExF) ∪ (ExG)</li> -<li>(E∪F) x G = (ExG) ∪ (FxG)</li> -<li>(E∪F) ∩ (GxH) = (E ∩ G) x (F ∩ H)</li> -<li>Generally speaking : (ExF) ∪ (GxH) ≠ (E∪G) x (F∪H)</li> +<li>ExF = ∅ ⇔ E=∅ OR F=∅<br /></li> +<li>ExF = FxE ⇔ E=F OR E=∅ OR F=∅<br /></li> +<li>E x (F∪G) = (ExF) ∪ (ExG)<br /></li> +<li>(E∪F) x G = (ExG) ∪ (FxG)<br /></li> +<li>(E∪F) ∩ (GxH) = (E ∩ G) x (F ∩ H)<br /></li> +<li>Generally speaking : (ExF) ∪ (GxH) ≠ (E∪G) x (F∪H)<br /></li> </ol> </div> </div> </div> </div> -<div id="outline-container-org8f809af" class="outline-2"> -<h2 id="org8f809af">Binary relations in a set :</h2> -<div class="outline-text-2" id="text-org8f809af"> +<div id="outline-container-org416c1e1" class="outline-2"> +<h2 id="org416c1e1">Binary relations in a set :</h2> +<div class="outline-text-2" id="text-org416c1e1"> </div> -<div id="outline-container-orgeb6cba6" class="outline-3"> -<h3 id="orgeb6cba6">Definition :</h3> -<div class="outline-text-3" id="text-orgeb6cba6"> +<div id="outline-container-org5ea795f" class="outline-3"> +<h3 id="org5ea795f">Definition :</h3> +<div class="outline-text-3" id="text-org5ea795f"> <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> +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-org1696189" class="outline-3"> -<h3 id="org1696189">Proprieties :</h3> -<div class="outline-text-3" id="text-org1696189"> +<div id="outline-container-orgb9d678f" class="outline-3"> +<h3 id="orgb9d678f">Proprieties :</h3> +<div class="outline-text-3" id="text-orgb9d678f"> <p> -Let E be a set and R a relation defined in E +Let E be a set and R a relation defined in E<br /> </p> <ol class="org-ol"> -<li>We say that R is reflexive if ∀ x ∈ E, xRx (for any element x in E,x is related to itself)</li> -<li>We say that R is symmetrical if ∀ x,y ∈ E , xRy ⇒ yRx</li> -<li>We say that R is transitive if ∀ x,y,z ∈ E (xRy , yRz) ⇒ xRz</li> -<li>We say that R is anti-symmetrical if ∀ x,y ∈ E xRy AND yRx ⇒ x = y</li> +<li>We say that R is reflexive if ∀ x ∈ E, xRx (for any element x in E,x is related to itself)<br /></li> +<li>We say that R is symmetrical if ∀ x,y ∈ E , xRy ⇒ yRx<br /></li> +<li>We say that R is transitive if ∀ x,y,z ∈ E (xRy , yRz) ⇒ xRz<br /></li> +<li>We say that R is anti-symmetrical if ∀ x,y ∈ E xRy AND yRx ⇒ x = y<br /></li> </ol> </div> </div> -<div id="outline-container-org38c5183" class="outline-3"> -<h3 id="org38c5183">Equivalence relationship :</h3> -<div class="outline-text-3" id="text-org38c5183"> +<div id="outline-container-org6df8952" class="outline-3"> +<h3 id="org6df8952">Equivalence relationship :</h3> +<div class="outline-text-3" id="text-org6df8952"> <p> -We say that R is a relation of equivalence in E if its reflexive, symetrical and transitive +We say that R is a relation of equivalence in E if its reflexive, symetrical and transitive<br /> </p> </div> -<div id="outline-container-org110e6fa" class="outline-4"> -<h4 id="org110e6fa">Equivalence class :</h4> -<div class="outline-text-4" id="text-org110e6fa"> +<div id="outline-container-orgfbd9232" class="outline-4"> +<h4 id="orgfbd9232">Equivalence class :</h4> +<div class="outline-text-4" id="text-orgfbd9232"> <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 : +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> <p> -<b>a̅ = {y ∈ E/ y R a}</b> +<b>a̅ = {y ∈ E/ y R a}</b><br /> </p> </div> <ul class="org-ul"> -<li><a id="org20e3b3b"></a>The quotient set :<br /> -<div class="outline-text-5" id="text-org20e3b3b"> +<li><a id="org1572848"></a>The quotient set :<br /> +<div class="outline-text-5" id="text-org1572848"> <p> -E/R = {̅a , a ∈ E} +E/R = {̅a , a ∈ E}<br /> </p> </div> </li> </ul> </div> </div> -<div id="outline-container-org25fec1b" class="outline-3"> -<h3 id="org25fec1b">Order relationship :</h3> -<div class="outline-text-3" id="text-org25fec1b"> +<div id="outline-container-org9a36dc1" class="outline-3"> +<h3 id="org9a36dc1">Order relationship :</h3> +<div class="outline-text-3" id="text-org9a36dc1"> <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. +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> <ol class="org-ol"> -<li>The order R is called total if ∀ x,y ∈ E xRy OR yRx</li> -<li>The order R is called partial if ∃ x,y ∈ E xR̅y AND yR̅x</li> +<li>The order R is called total if ∀ x,y ∈ E xRy OR yRx<br /></li> +<li>The order R is called partial if ∃ x,y ∈ E xR̅y AND yR̅x<br /></li> </ol> </div> -<div id="outline-container-orgc094acc" class="outline-4"> -<h4 id="orgc094acc"><span class="todo TODO">TODO</span> Examples :</h4> -<div class="outline-text-4" id="text-orgc094acc"> +<div id="outline-container-orgb496cba" class="outline-4"> +<h4 id="orgb496cba"><span class="todo TODO">TODO</span> Examples :</h4> +<div class="outline-text-4" id="text-orgb496cba"> <p> -∀x,y ∈ ℝ , xRy ⇔ x²-y²=x-y +∀x,y ∈ ℝ , xRy ⇔ x²-y²=x-y<br /> </p> <ol class="org-ol"> -<li>Prove that R is an equivalence relation</li> -<li>Let a ∈ ℝ, find ̅a</li> +<li>Prove that R is an equivalence relation<br /></li> +<li>Let a ∈ ℝ, find ̅a<br /></li> </ol> </div> </div> </div> </div> +<div id="outline-container-org54d5489" class="outline-2"> +<h2 id="org54d5489">TP exercices <i>Oct 20</i> :</h2> +<div class="outline-text-2" id="text-org54d5489"> +</div> +<div id="outline-container-orgdfd55ca" class="outline-3"> +<h3 id="orgdfd55ca">Exercice 3 :</h3> +<div class="outline-text-3" id="text-orgdfd55ca"> +</div> +<div id="outline-container-org4100fe3" class="outline-4"> +<h4 id="org4100fe3">Question 3</h4> +<div class="outline-text-4" id="text-org4100fe3"> +<p> +Montrer par l’absurde que P : ∀x ∈ ℝ*, √(4+x³) ≠ 2 + x³/4 est vraies<br /> +</p> + +<p class="verse"> +On suppose que ∃ x ∈ ℝ* , √(4+x³) = 2 + x³/4<br /> +4+x³ = (2 + x³/4)²<br /> +4+x³ = 4 + x⁶/16 + 4*(x³/4)<br /> +4+x³ = 4 + x⁶/16 + x³<br /> +x⁶/16 = 0<br /> +x⁶ = 0<br /> +x = 0 . Or, x appartiens a ℝ\{0}, donc P̅ est fausse. Ce qui est equivalent a dire que P est vraie<br /> +</p> +</div> +</div> +</div> +<div id="outline-container-org019b5e0" class="outline-3"> +<h3 id="org019b5e0">Exercice 4 :</h3> +<div class="outline-text-3" id="text-org019b5e0"> +</div> +<div id="outline-container-org2ae1181" class="outline-4"> +<h4 id="org2ae1181"><span class="done DONE">DONE</span> Question 1 :</h4> +<div class="outline-text-4" id="text-org2ae1181"> +<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 /> +1. <b>On vérifie P(n) pour n = 1</b><br /> +(1 ,k=1)Σ1/k(k+1) = 1/1(1+1)<br /> +                  = 1/2 — (1)<br /> +1 - 1/1+1 = 1 - 1/2<br /> +                  = 1/2 — (2)<br /> +De (1) et (2), P(0) est vraie -— (a)<br /> +<br /> +2. <b>On suppose que P(n) est vraie pour n ≥ n1 puis on vérifie pour n+1</b><br /> +(n ,k=1)Σ1/k(k+1) = 1 - 1/1+n<br /> +(n ,k=1)Σ1/k(k+1) + 1/(n+1)(n+2) = 1 - (1/(1+n)) + 1/(n+1)(n+2)<br /> +(n+1 ,k=1)Σ1/k(k+1) = 1 - 1/(n+1) + 1/[(n+1)(n+2)]<br /> +(n+1 ,k=1)Σ1/k(k+1) = 1 + 1/[(n+1)(n+2)] - (n+2)/[(n+1)(n+2)]<br /> +(n+1 ,k=1)Σ1/k(k+1) = 1 + [1-(n+2)]/[(n+1)(n+2)]<br /> +(n+1 ,k=1)Σ1/k(k+1) = 1 + [-n-1]/[(n+1)(n+2)]<br /> +(n+1 ,k=1)Σ1/k(k+1) = 1 - [n+1]/[(n+1)(n+2)]<br /> +(n+1 ,k=1)Σ1/k(k+1) = 1 - 1/(n+1+1) <b>CQFD</b><br /> +<br /> +Donc P(n+1) est vraie. -— (b)<br /> +De (a) et (b) on conclus que la proposition de départ est vraie<br /> +</p> +</div> +</div> +</div> +</div> </div> <div id="postamble" class="status"> <p class="author">Author: Crystal</p> -<p class="date">Created: 2023-10-17 Tue 22:32</p> +<p class="date">Created: 2023-10-20 Fri 15:12</p> </div> </body> </html> \ No newline at end of file |