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diff --git a/uni_notes/algebra.html b/uni_notes/algebra.html index fd6602c..c42a1b5 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> -<!-- 2023-10-20 Fri 15:12 --> +<!-- 2023-10-23 Mon 19:39 --> <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> @@ -23,132 +23,161 @@ <h2>Table of Contents</h2> <div id="text-table-of-contents" role="doc-toc"> <ul> -<li><a href="#orgdb6601b">Contenu de la Matiére</a> +<li><a href="#org4a250bc">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> +<li><a href="#orga898532">Rappels et compléments (11H)</a></li> +<li><a href="#org01bcbce">Structures Algébriques (11H)</a></li> +<li><a href="#orgbe1e218">Polynômes et fractions rationnelles</a></li> </ul> </li> -<li><a href="#org8356dfb">Premier cours : Logique mathématique et méthodes du raisonnement mathématique <i>Sep 25</i> :</a> +<li><a href="#orgc75277f">Premier cours : Logique mathématique et méthodes du raisonnement mathématique <i>Sep 25</i> :</a> <ul> -<li><a href="#org4521340">Properties:</a> +<li><a href="#orge4d9578">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> +<li><a href="#orgb979f5e"><b>Absorption</b>:</a></li> +<li><a href="#org747a426"><b>Commutativity</b>:</a></li> +<li><a href="#org515acb7"><b>Associativity</b>:</a></li> +<li><a href="#orgbd31315"><b>Distributivity</b>:</a></li> +<li><a href="#org6ffa08f"><b>Neutral element</b>:</a></li> +<li><a href="#org5687242"><b>Negation of a conjunction & a disjunction</b>:</a></li> +<li><a href="#orge1d09c1"><b>Transitivity</b>:</a></li> +<li><a href="#orgcbc82d0"><b>Contraposition</b>:</a></li> +<li><a href="#org41b3b67">God only knows what this property is called:</a></li> </ul> </li> -<li><a href="#org4210e18">Some exercices I found online :</a> +<li><a href="#org8339e2b">Some exercices I found online :</a> <ul> -<li><a href="#org46d1ee6">USTHB 2022/2023 Section B :</a></li> +<li><a href="#org4e164f4">USTHB 2022/2023 Section B :</a></li> </ul> </li> </ul> </li> -<li><a href="#org732b9dd">2éme cours <i>Oct 2</i></a> +<li><a href="#org966520a">2éme cours <i>Oct 2</i></a> <ul> -<li><a href="#orgdfba00a">Quantifiers</a> +<li><a href="#orgafe4a7b">Quantifiers</a> <ul> -<li><a href="#org93d5891">Proprieties</a></li> +<li><a href="#org5441c86">Proprieties</a></li> </ul> </li> -<li><a href="#orgf21a239">Multi-parameter proprieties :</a></li> -<li><a href="#org779f917">Methods of mathematical reasoning :</a> +<li><a href="#orga224095">Multi-parameter proprieties :</a></li> +<li><a href="#org019a7b1">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> +<li><a href="#org235fea9">Direct reasoning :</a></li> +<li><a href="#orgae58c37">Reasoning by the Absurd:</a></li> +<li><a href="#orgb7e7f5f">Reasoning by contraposition:</a></li> +<li><a href="#org2199fd0">Reasoning by counter example:</a></li> </ul> </li> </ul> </li> -<li><a href="#orgfddd579">3eme Cours : <i>Oct 9</i></a> +<li><a href="#org97d10cc">3eme Cours : <i>Oct 9</i></a> <ul> <li> <ul> -<li><a href="#org21eab57">Reasoning by recurrence :</a></li> +<li><a href="#orgc324cd9">Reasoning by recurrence :</a></li> </ul> </li> </ul> </li> -<li><a href="#org155c9a9">4eme Cours : Chapitre 2 : Sets and Operations</a> +<li><a href="#org924a736">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> +<li><a href="#org9d2cb2d">Definition of a set :</a></li> +<li><a href="#org3b107a4">Belonging, inclusion, and equality :</a></li> +<li><a href="#orgbba2f2c">Intersections and reunions :</a> <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> +<li><a href="#org4fd2e61">Intersection:</a></li> +<li><a href="#orgabd991c">Union:</a></li> +<li><a href="#org083102e">Difference between two sets:</a></li> +<li><a href="#orgcc1969e">Complimentary set:</a></li> +<li><a href="#org72511ff">Symmetrical difference</a></li> </ul> </li> -<li><a href="#org1e0c21a">Proprieties :</a> +<li><a href="#org44b1b96">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> +<li><a href="#orga3eac79">Commutativity:</a></li> +<li><a href="#org1a9121a">Associativity:</a></li> +<li><a href="#orgaf7fe3b">Distributivity:</a></li> +<li><a href="#org658b728">Lois de Morgan:</a></li> +<li><a href="#org4f0dd58">An other one:</a></li> +<li><a href="#orgdd60033">An other one:</a></li> +<li><a href="#orgbf5feb1">And an other one:</a></li> +<li><a href="#orgefabd47">And the last one:</a></li> </ul> </li> </ul> </li> -<li><a href="#orgeb2675e">5eme cours: L’ensemble des parties d’un ensemble <i>Oct 16</i></a> +<li><a href="#orgc9ed06c">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> +<li><a href="#org7a1da3c">Notes :</a></li> +<li><a href="#orga6e5f8a">Examples :</a></li> </ul> </li> -<li><a href="#org3da3de1">Partition of a set :</a></li> -<li><a href="#org077b994">Cartesian products :</a> +<li><a href="#org7286ec5">Partition of a set :</a></li> +<li><a href="#orgd8e00ac">Cartesian products :</a> <ul> -<li><a href="#org72822f5">Example :</a></li> -<li><a href="#org04e1be3">Some proprieties:</a></li> +<li><a href="#orgdb491ee">Example :</a></li> +<li><a href="#org28c23b2">Some proprieties:</a></li> </ul> </li> </ul> </li> -<li><a href="#org416c1e1">Binary relations in a set :</a> +<li><a href="#orgfdbe4f3">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> +<li><a href="#org3696656">Definition :</a></li> +<li><a href="#orgd32f673">Proprieties :</a></li> +<li><a href="#org22f460a">Equivalence relationship :</a> <ul> -<li><a href="#orgfbd9232">Equivalence class :</a></li> +<li><a href="#org68ddde2">Equivalence class :</a></li> </ul> </li> -<li><a href="#org9a36dc1">Order relationship :</a> +<li><a href="#orge976c7e">Order relationship :</a> <ul> -<li><a href="#orgb496cba"><span class="todo TODO">TODO</span> Examples :</a></li> +<li><a href="#org1f19847"><span class="todo TODO">TODO</span> Examples :</a></li> </ul> </li> </ul> </li> -<li><a href="#org54d5489">TP exercices <i>Oct 20</i> :</a> +<li><a href="#orgc956a73">TP exercices <i>Oct 20</i> :</a> <ul> -<li><a href="#orgdfd55ca">Exercice 3 :</a> +<li><a href="#org15b0e75">Exercice 3 :</a> <ul> -<li><a href="#org4100fe3">Question 3</a></li> +<li><a href="#orgb132892">Question 3</a></li> </ul> </li> -<li><a href="#org019b5e0">Exercice 4 :</a> +<li><a href="#org9a4006b">Exercice 4 :</a> <ul> -<li><a href="#org2ae1181"><span class="done DONE">DONE</span> Question 1 :</a></li> +<li><a href="#org43cf6d6"><span class="done DONE">DONE</span> Question 1 :</a></li> +</ul> +</li> +</ul> +</li> +<li><a href="#org429ab91">Chapter 3 : Applications</a> +<ul> +<li><a href="#org4a4b3cf">3.1 Generalities about applications :</a> +<ul> +<li><a href="#org0cac0c6">Definition :</a></li> +<li><a href="#orgc048e93">Restriction and prolongation of an application :</a></li> +<li><a href="#org8e61361">Composition of applications :</a></li> +</ul> +</li> +<li><a href="#org5c096db">3.2 Injection, surjection and bijection :</a> +<ul> +<li><a href="#org4162b56">Proposition :</a></li> +</ul> +</li> +<li><a href="#org736de6c">3.3 Reciprocal applications :</a> +<ul> +<li><a href="#orgafb7f85">Def :</a></li> +<li><a href="#orgec3a3d6">Theorem :</a></li> +<li><a href="#org940e2d6">Some proprieties :</a></li> +</ul> +</li> +<li><a href="#org2d173c2">3.4 Direct Image and reciprocal Image :</a> +<ul> +<li><a href="#org769c809">Direct Image :</a></li> +<li><a href="#org7d705d3">Reciprocal image :</a></li> </ul> </li> </ul> @@ -156,13 +185,13 @@ </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 id="outline-container-org4a250bc" class="outline-2"> +<h2 id="org4a250bc">Contenu de la Matiére</h2> +<div class="outline-text-2" id="text-org4a250bc"> </div> -<div id="outline-container-orgc89165c" class="outline-3"> -<h3 id="orgc89165c">Rappels et compléments (11H)</h3> -<div class="outline-text-3" id="text-orgc89165c"> +<div id="outline-container-orga898532" class="outline-3"> +<h3 id="orga898532">Rappels et compléments (11H)</h3> +<div class="outline-text-3" id="text-orga898532"> <ul class="org-ul"> <li>Logique mathématique et méthodes du raisonnement mathématique<br /></li> <li>Ensembles et Relations<br /></li> @@ -170,9 +199,9 @@ </ul> </div> </div> -<div id="outline-container-org1ae59da" class="outline-3"> -<h3 id="org1ae59da">Structures Algébriques (11H)</h3> -<div class="outline-text-3" id="text-org1ae59da"> +<div id="outline-container-org01bcbce" class="outline-3"> +<h3 id="org01bcbce">Structures Algébriques (11H)</h3> +<div class="outline-text-3" id="text-org01bcbce"> <ul class="org-ul"> <li>Groupes et morphisme de groupes<br /></li> <li>Anneaux et morphisme d’anneaux<br /></li> @@ -180,9 +209,9 @@ </ul> </div> </div> -<div id="outline-container-orgdcb51e0" class="outline-3"> -<h3 id="orgdcb51e0">Polynômes et fractions rationnelles</h3> -<div class="outline-text-3" id="text-orgdcb51e0"> +<div id="outline-container-orgbe1e218" class="outline-3"> +<h3 id="orgbe1e218">Polynômes et fractions rationnelles</h3> +<div class="outline-text-3" id="text-orgbe1e218"> <ul class="org-ul"> <li>Notion du polynôme à une indéterminée á coefficients dans un anneau<br /></li> <li>Opérations Algébriques sur les polynômes<br /></li> @@ -195,9 +224,9 @@ </div> </div> </div> -<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"> +<div id="outline-container-orgc75277f" class="outline-2"> +<h2 id="orgc75277f">Premier cours : Logique mathématique et méthodes du raisonnement mathématique <i>Sep 25</i> :</h2> +<div class="outline-text-2" id="text-orgc75277f"> <p> Let <b>P</b> <b>Q</b> and <b>R</b> be propositions which can either be <b>True</b> or <b>False</b>. And let’s also give the value <b>1</b> to each <b>True</b> proposition and <b>0</b> to each false one.<br /> </p> @@ -547,13 +576,13 @@ A proposition is equivalent to another only when both of them have <b>the same v <i>Note: P implying Q is equivalent to P̅ implying Q̅, or: (P ⇒ Q) ⇔ (P̅ ⇒ Q̅)</i><br /> </p> </div> -<div id="outline-container-org4521340" class="outline-3"> -<h3 id="org4521340">Properties:</h3> -<div class="outline-text-3" id="text-org4521340"> +<div id="outline-container-orge4d9578" class="outline-3"> +<h3 id="orge4d9578">Properties:</h3> +<div class="outline-text-3" id="text-orge4d9578"> </div> -<div id="outline-container-org84e1ec3" class="outline-4"> -<h4 id="org84e1ec3"><b>Absorption</b>:</h4> -<div class="outline-text-4" id="text-org84e1ec3"> +<div id="outline-container-orgb979f5e" class="outline-4"> +<h4 id="orgb979f5e"><b>Absorption</b>:</h4> +<div class="outline-text-4" id="text-orgb979f5e"> <p> (P ∨ P) ⇔ P<br /> </p> @@ -563,9 +592,9 @@ A proposition is equivalent to another only when both of them have <b>the same v </p> </div> </div> -<div id="outline-container-org2345992" class="outline-4"> -<h4 id="org2345992"><b>Commutativity</b>:</h4> -<div class="outline-text-4" id="text-org2345992"> +<div id="outline-container-org747a426" class="outline-4"> +<h4 id="org747a426"><b>Commutativity</b>:</h4> +<div class="outline-text-4" id="text-org747a426"> <p> (P ∧ Q) ⇔ (Q ∧ P)<br /> </p> @@ -575,9 +604,9 @@ A proposition is equivalent to another only when both of them have <b>the same v </p> </div> </div> -<div id="outline-container-orgfee557e" class="outline-4"> -<h4 id="orgfee557e"><b>Associativity</b>:</h4> -<div class="outline-text-4" id="text-orgfee557e"> +<div id="outline-container-org515acb7" class="outline-4"> +<h4 id="org515acb7"><b>Associativity</b>:</h4> +<div class="outline-text-4" id="text-org515acb7"> <p> P ∧ (Q ∧ R) ⇔ (P ∧ Q) ∧ R<br /> </p> @@ -587,9 +616,9 @@ P ∨ (Q ∨ R) ⇔ (P ∨ Q) ∨ R<br /> </p> </div> </div> -<div id="outline-container-orgb55cfb2" class="outline-4"> -<h4 id="orgb55cfb2"><b>Distributivity</b>:</h4> -<div class="outline-text-4" id="text-orgb55cfb2"> +<div id="outline-container-orgbd31315" class="outline-4"> +<h4 id="orgbd31315"><b>Distributivity</b>:</h4> +<div class="outline-text-4" id="text-orgbd31315"> <p> P ∧ (Q ∨ R) ⇔ (P ∧ Q) ∨ (P ∧ R)<br /> </p> @@ -599,9 +628,9 @@ P ∨ (Q ∧ R) ⇔ (P ∨ Q) ∧ (P ∨ R)<br /> </p> </div> </div> -<div id="outline-container-orgb0de2bb" class="outline-4"> -<h4 id="orgb0de2bb"><b>Neutral element</b>:</h4> -<div class="outline-text-4" id="text-orgb0de2bb"> +<div id="outline-container-org6ffa08f" class="outline-4"> +<h4 id="org6ffa08f"><b>Neutral element</b>:</h4> +<div class="outline-text-4" id="text-org6ffa08f"> <p> <i>We define proposition <b>T</b> to be always <b>true</b> and <b>F</b> to be always <b>false</b></i><br /> </p> @@ -615,9 +644,9 @@ P ∨ F ⇔ P<br /> </p> </div> </div> -<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"> +<div id="outline-container-org5687242" class="outline-4"> +<h4 id="org5687242"><b>Negation of a conjunction & a disjunction</b>:</h4> +<div class="outline-text-4" id="text-org5687242"> <p> Now we won’t use bars here because my lazy ass doesn’t know how, so instead I will use not()!!!<br /> </p> @@ -635,25 +664,25 @@ not(<b>P ∨ Q</b>) ⇔ P̅ ∧ Q̅<br /> </p> </div> </div> -<div id="outline-container-org4e29124" class="outline-4"> -<h4 id="org4e29124"><b>Transitivity</b>:</h4> -<div class="outline-text-4" id="text-org4e29124"> +<div id="outline-container-orge1d09c1" class="outline-4"> +<h4 id="orge1d09c1"><b>Transitivity</b>:</h4> +<div class="outline-text-4" id="text-orge1d09c1"> <p> [(P ⇒ Q) AND (Q ⇒ R)] ⇔ P ⇒ R<br /> </p> </div> </div> -<div id="outline-container-org520b7b0" class="outline-4"> -<h4 id="org520b7b0"><b>Contraposition</b>:</h4> -<div class="outline-text-4" id="text-org520b7b0"> +<div id="outline-container-orgcbc82d0" class="outline-4"> +<h4 id="orgcbc82d0"><b>Contraposition</b>:</h4> +<div class="outline-text-4" id="text-orgcbc82d0"> <p> (P ⇒ Q) ⇔ (Q̅ ⇒ P̅)<br /> </p> </div> </div> -<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"> +<div id="outline-container-org41b3b67" class="outline-4"> +<h4 id="org41b3b67">God only knows what this property is called:</h4> +<div class="outline-text-4" id="text-org41b3b67"> <p> <i>If</i><br /> </p> @@ -680,17 +709,17 @@ Q is always true<br /> </div> </div> </div> -<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 id="outline-container-org8339e2b" class="outline-3"> +<h3 id="org8339e2b">Some exercices I found online :</h3> +<div class="outline-text-3" id="text-org8339e2b"> </div> -<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 id="outline-container-org4e164f4" class="outline-4"> +<h4 id="org4e164f4">USTHB 2022/2023 Section B :</h4> +<div class="outline-text-4" id="text-org4e164f4"> </div> <ul class="org-ul"> -<li><a id="orga6d3248"></a>Exercice 1: Démontrer les équivalences suivantes:<br /> -<div class="outline-text-5" id="text-orga6d3248"> +<li><a id="org1202cc7"></a>Exercice 1: Démontrer les équivalences suivantes:<br /> +<div class="outline-text-5" id="text-org1202cc7"> <ol class="org-ol"> <li><p> (P ⇒ Q) ⇔ (Q̅ ⇒ P̅)<br /> @@ -744,8 +773,8 @@ Literally the same as above 🩷<br /> </ol> </div> </li> -<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"> +<li><a id="org993b830"></a>Exercice 2: Dire si les propositions suivantes sont vraies ou fausses, et les nier:<br /> +<div class="outline-text-5" id="text-org993b830"> <ol class="org-ol"> <li><p> ∀x ∈ ℝ ,∃y ∈ ℝ*+, tels que e^x = y<br /> @@ -878,13 +907,13 @@ y + x < 8<br /> </div> </div> </div> -<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 id="outline-container-org966520a" class="outline-2"> +<h2 id="org966520a">2éme cours <i>Oct 2</i></h2> +<div class="outline-text-2" id="text-org966520a"> </div> -<div id="outline-container-orgdfba00a" class="outline-3"> -<h3 id="orgdfba00a">Quantifiers</h3> -<div class="outline-text-3" id="text-orgdfba00a"> +<div id="outline-container-orgafe4a7b" class="outline-3"> +<h3 id="orgafe4a7b">Quantifiers</h3> +<div class="outline-text-3" id="text-orgafe4a7b"> <p> A propriety P can depend on a parameter x<br /> </p> @@ -900,8 +929,8 @@ A propriety P can depend on a parameter x<br /> </p> </div> <ul class="org-ul"> -<li><a id="org9a107f8"></a>Example<br /> -<div class="outline-text-6" id="text-org9a107f8"> +<li><a id="org11a2eef"></a>Example<br /> +<div class="outline-text-6" id="text-org11a2eef"> <p> P(x) : x+1≥0<br /> </p> @@ -912,13 +941,13 @@ P(X) is True or False depending on the values of x<br /> </div> </li> </ul> -<div id="outline-container-org93d5891" class="outline-4"> -<h4 id="org93d5891">Proprieties</h4> -<div class="outline-text-4" id="text-org93d5891"> +<div id="outline-container-org5441c86" class="outline-4"> +<h4 id="org5441c86">Proprieties</h4> +<div class="outline-text-4" id="text-org5441c86"> </div> <ul class="org-ul"> -<li><a id="orgcc6c2bd"></a>Propriety Number 1:<br /> -<div class="outline-text-5" id="text-orgcc6c2bd"> +<li><a id="org92d20b7"></a>Propriety Number 1:<br /> +<div class="outline-text-5" id="text-org92d20b7"> <p> The negation of the universal quantifier is the existential quantifier, and vice-versa :<br /> </p> @@ -929,8 +958,8 @@ The negation of the universal quantifier is the existential quantifier, and vice </ul> </div> <ul class="org-ul"> -<li><a id="org4d384a3"></a>Example:<br /> -<div class="outline-text-6" id="text-org4d384a3"> +<li><a id="orgd155b7d"></a>Example:<br /> +<div class="outline-text-6" id="text-orgd155b7d"> <p> ∀ x ≥ 1 x² > 5 ⇔ ∃ x ≥ 1 x² < 5<br /> </p> @@ -938,8 +967,8 @@ The negation of the universal quantifier is the existential quantifier, and vice </li> </ul> </li> -<li><a id="org372a28e"></a>Propriety Number 2:<br /> -<div class="outline-text-5" id="text-org372a28e"> +<li><a id="orga5c524a"></a>Propriety Number 2:<br /> +<div class="outline-text-5" id="text-orga5c524a"> <p> <b>∀x ∈ E, [P(x) ∧ Q(x)] ⇔ [∀ x ∈ E, P(x)] ∧ [∀ x ∈ E, Q(x)]</b><br /> </p> @@ -950,8 +979,8 @@ The propriety “For any value of x from a set E , P(x) and Q(x)” is e </p> </div> <ul class="org-ul"> -<li><a id="org7649498"></a>Example :<br /> -<div class="outline-text-6" id="text-org7649498"> +<li><a id="orgfbd033f"></a>Example :<br /> +<div class="outline-text-6" id="text-orgfbd033f"> <p> P(x) : sqrt(x) > 0 ; Q(x) : x ≥ 1<br /> </p> @@ -969,8 +998,8 @@ P(x) : sqrt(x) > 0 ; Q(x) : x ≥ 1<br /> </li> </ul> </li> -<li><a id="orgbccb33f"></a>Propriety Number 3:<br /> -<div class="outline-text-5" id="text-orgbccb33f"> +<li><a id="orgf25f95e"></a>Propriety Number 3:<br /> +<div class="outline-text-5" id="text-orgf25f95e"> <p> <b>∃ x ∈ E, [P(x) ∧ Q(x)] <i>⇒</i> [∃ x ∈ E, P(x)] ∧ [∃ x ∈ E, Q(x)]</b><br /> </p> @@ -981,8 +1010,8 @@ P(x) : sqrt(x) > 0 ; Q(x) : x ≥ 1<br /> </p> </div> <ul class="org-ul"> -<li><a id="orgf623821"></a>Example of why it’s NOT an equivalence :<br /> -<div class="outline-text-6" id="text-orgf623821"> +<li><a id="org1eb9ad5"></a>Example of why it’s NOT an equivalence :<br /> +<div class="outline-text-6" id="text-org1eb9ad5"> <p> P(x) : x > 5 ; Q(x) : x < 5<br /> </p> @@ -995,8 +1024,8 @@ Of course there is no value of x such as its inferior and superior to 5 at the s </li> </ul> </li> -<li><a id="orgda152b2"></a>Propriety Number 4:<br /> -<div class="outline-text-5" id="text-orgda152b2"> +<li><a id="org1b17eb0"></a>Propriety Number 4:<br /> +<div class="outline-text-5" id="text-org1b17eb0"> <p> <b>[∀ x ∈ E, P(x)] ∨ [∀ x ∈ E, Q(x)] <i>⇒</i> ∀x ∈ E, [P(x) ∨ Q(x)]</b><br /> </p> @@ -1010,16 +1039,16 @@ Of course there is no value of x such as its inferior and superior to 5 at the s </ul> </div> </div> -<div id="outline-container-orgf21a239" class="outline-3"> -<h3 id="orgf21a239">Multi-parameter proprieties :</h3> -<div class="outline-text-3" id="text-orgf21a239"> +<div id="outline-container-orga224095" class="outline-3"> +<h3 id="orga224095">Multi-parameter proprieties :</h3> +<div class="outline-text-3" id="text-orga224095"> <p> A propriety P can depend on two or more parameters, for convenience we call them x,y,z…etc<br /> </p> </div> <ul class="org-ul"> -<li><a id="orgdda5feb"></a>Example :<br /> -<div class="outline-text-6" id="text-orgdda5feb"> +<li><a id="orgf04d41f"></a>Example :<br /> +<div class="outline-text-6" id="text-orgf04d41f"> <p> P(x,y): x+y > 0<br /> </p> @@ -1035,8 +1064,8 @@ P(-2,-1) is a False one<br /> </p> </div> </li> -<li><a id="org08d728b"></a>WARNING :<br /> -<div class="outline-text-6" id="text-org08d728b"> +<li><a id="orgb4df659"></a>WARNING :<br /> +<div class="outline-text-6" id="text-orgb4df659"> <p> ∀x ∈ E, ∃y ∈ F , P(x,y)<br /> </p> @@ -1052,8 +1081,8 @@ Are different because in the first one y depends on x, while in the second one, </p> </div> <ul class="org-ul"> -<li><a id="org78dcf22"></a>Example :<br /> -<div class="outline-text-7" id="text-org78dcf22"> +<li><a id="orge473c22"></a>Example :<br /> +<div class="outline-text-7" id="text-orge473c22"> <p> ∀ x ∈ ℕ , ∃ y ∈ ℕ y > x -–— True<br /> </p> @@ -1067,8 +1096,8 @@ Are different because in the first one y depends on x, while in the second one, </ul> </li> </ul> -<li><a id="org088c862"></a>Proprieties :<br /> -<div class="outline-text-5" id="text-org088c862"> +<li><a id="org5f7adfc"></a>Proprieties :<br /> +<div class="outline-text-5" id="text-org5f7adfc"> <ol class="org-ol"> <li>not(∀x ∈ E ,∃y ∈ F P(x,y)) ⇔ ∃x ∈ E, ∀y ∈ F not(P(x,y))<br /></li> <li>not(∃x ∈ E ,∀y ∈ F P(x,y)) ⇔ ∀x ∈ E, ∃y ∈ F not(P(x,y))<br /></li> @@ -1077,20 +1106,20 @@ Are different because in the first one y depends on x, while in the second one, </li> </ul> </div> -<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 id="outline-container-org019a7b1" class="outline-3"> +<h3 id="org019a7b1">Methods of mathematical reasoning :</h3> +<div class="outline-text-3" id="text-org019a7b1"> </div> -<div id="outline-container-orgc6832c3" class="outline-4"> -<h4 id="orgc6832c3">Direct reasoning :</h4> -<div class="outline-text-4" id="text-orgc6832c3"> +<div id="outline-container-org235fea9" class="outline-4"> +<h4 id="org235fea9">Direct reasoning :</h4> +<div class="outline-text-4" id="text-org235fea9"> <p> To show that an implication P ⇒ Q is true, we suppose that P is true and we show that Q is true<br /> </p> </div> <ul class="org-ul"> -<li><a id="org8dbaab8"></a>Example:<br /> -<div class="outline-text-5" id="text-org8dbaab8"> +<li><a id="orgced11f5"></a>Example:<br /> +<div class="outline-text-5" id="text-orgced11f5"> <p> Let a,b be two Real numbers, we have to prove that <b>a² + b² = 1 ⇒ |a + b| ≤ 2</b><br /> </p> @@ -1133,9 +1162,9 @@ a²+b²=1 ⇒ |a + b| ≤ 2 <b>Which is what we wanted to prove, therefor the im </li> </ul> </div> -<div id="outline-container-orgf140b16" class="outline-4"> -<h4 id="orgf140b16">Reasoning by the Absurd:</h4> -<div class="outline-text-4" id="text-orgf140b16"> +<div id="outline-container-orgae58c37" class="outline-4"> +<h4 id="orgae58c37">Reasoning by the Absurd:</h4> +<div class="outline-text-4" id="text-orgae58c37"> <p> To prove that a proposition is True, we suppose that it’s False and we must come to a contradiction<br /> </p> @@ -1146,8 +1175,8 @@ And to prove that an implication P ⇒ Q is true using the reasoning by the absu </p> </div> <ul class="org-ul"> -<li><a id="org1fefbfb"></a>Example:<br /> -<div class="outline-text-5" id="text-org1fefbfb"> +<li><a id="org8dc4906"></a>Example:<br /> +<div class="outline-text-5" id="text-org8dc4906"> <p> Prove that this proposition is correct using the reasoning by the absurd : ∀x ∈ ℝ* , sqrt(1+x²) ≠ 1 + x²/2<br /> </p> @@ -1165,17 +1194,17 @@ sqrt(1+x²) = 1 + x²/2 ; 1 + x² = (1+x²/2)² ; 1 + x² = 1 + x^4/4 + x² ; </li> </ul> </div> -<div id="outline-container-org320dc57" class="outline-4"> -<h4 id="org320dc57">Reasoning by contraposition:</h4> -<div class="outline-text-4" id="text-org320dc57"> +<div id="outline-container-orgb7e7f5f" class="outline-4"> +<h4 id="orgb7e7f5f">Reasoning by contraposition:</h4> +<div class="outline-text-4" id="text-orgb7e7f5f"> <p> If an implication P ⇒ Q is too hard to prove, we just have to prove not(Q) ⇒ not(P) is true !!! or in other words that both not(P) and not(Q) are true<br /> </p> </div> </div> -<div id="outline-container-org27943b2" class="outline-4"> -<h4 id="org27943b2">Reasoning by counter example:</h4> -<div class="outline-text-4" id="text-org27943b2"> +<div id="outline-container-org2199fd0" class="outline-4"> +<h4 id="org2199fd0">Reasoning by counter example:</h4> +<div class="outline-text-4" id="text-org2199fd0"> <p> To prove that a proposition ∀x ∈ E, P(x) is false, all we have to do is find a single value of x from E such as not(P(x)) is true<br /> </p> @@ -1183,20 +1212,20 @@ To prove that a proposition ∀x ∈ E, P(x) is false, all we have to do is find </div> </div> </div> -<div id="outline-container-orgfddd579" class="outline-2"> -<h2 id="orgfddd579">3eme Cours : <i>Oct 9</i></h2> -<div class="outline-text-2" id="text-orgfddd579"> +<div id="outline-container-org97d10cc" class="outline-2"> +<h2 id="org97d10cc">3eme Cours : <i>Oct 9</i></h2> +<div class="outline-text-2" id="text-org97d10cc"> </div> -<div id="outline-container-org21eab57" class="outline-4"> -<h4 id="org21eab57">Reasoning by recurrence :</h4> -<div class="outline-text-4" id="text-org21eab57"> +<div id="outline-container-orgc324cd9" class="outline-4"> +<h4 id="orgc324cd9">Reasoning by recurrence :</h4> +<div class="outline-text-4" id="text-orgc324cd9"> <p> P is a propriety dependent of <b>n ∈ ℕ</b>. If for n0 ∈ ℕ P(n0) is true, and if for n ≥ n0 (P(n) ⇒ P(n+1)) is true. Then P(n) is true for n ≥ n0<br /> </p> </div> <ul class="org-ul"> -<li><a id="org2a3aa46"></a>Example:<br /> -<div class="outline-text-5" id="text-org2a3aa46"> +<li><a id="org00a2b7b"></a>Example:<br /> +<div class="outline-text-5" id="text-org00a2b7b"> <p> Let’s prove that ∀ n ≥ 1 , (n,k=1)Σk = [n(n+1)]/2<br /> </p> @@ -1232,21 +1261,21 @@ For n ≥ 1. We assume that P(n) is true, OR : <b>(n, k=1)Σk = n(n+1)/2</b>. We </ul> </div> </div> -<div id="outline-container-org155c9a9" class="outline-2"> -<h2 id="org155c9a9">4eme Cours : Chapitre 2 : Sets and Operations</h2> -<div class="outline-text-2" id="text-org155c9a9"> +<div id="outline-container-org924a736" class="outline-2"> +<h2 id="org924a736">4eme Cours : Chapitre 2 : Sets and Operations</h2> +<div class="outline-text-2" id="text-org924a736"> </div> -<div id="outline-container-org5bb0b39" class="outline-3"> -<h3 id="org5bb0b39">Definition of a set :</h3> -<div class="outline-text-3" id="text-org5bb0b39"> +<div id="outline-container-org9d2cb2d" class="outline-3"> +<h3 id="org9d2cb2d">Definition of a set :</h3> +<div class="outline-text-3" id="text-org9d2cb2d"> <p> A set is a collection of objects that share the sane propriety<br /> </p> </div> </div> -<div id="outline-container-org469078a" class="outline-3"> -<h3 id="org469078a">Belonging, inclusion, and equality :</h3> -<div class="outline-text-3" id="text-org469078a"> +<div id="outline-container-org3b107a4" class="outline-3"> +<h3 id="org3b107a4">Belonging, inclusion, and equality :</h3> +<div class="outline-text-3" id="text-org3b107a4"> <ol class="org-ol"> <li>Let E be a set. If x is an element of E, we say that x belongs to E we write <b>x ∈ E</b>, and if it doesn’t, we write <b>x ∉ E</b><br /></li> <li>A set E is included in a set F if all elements of E are elements of F and we write <b>E ⊂ F ⇔ (∀x , x ∈ E ⇒ x ∈ F)</b>. We say that E is a subset of F, or a part of F. The negation of this propriety is : <b>E ⊄ F ⇔ ∃x , x ∈ E and x ⊄ F</b><br /></li> @@ -1255,13 +1284,13 @@ A set is a collection of objects that share the sane propriety<br /> </ol> </div> </div> -<div id="outline-container-org1461e60" class="outline-3"> -<h3 id="org1461e60">Intersections and reunions :</h3> -<div class="outline-text-3" id="text-org1461e60"> +<div id="outline-container-orgbba2f2c" class="outline-3"> +<h3 id="orgbba2f2c">Intersections and reunions :</h3> +<div class="outline-text-3" id="text-orgbba2f2c"> </div> -<div id="outline-container-orgd9db499" class="outline-4"> -<h4 id="orgd9db499">Intersection:</h4> -<div class="outline-text-4" id="text-orgd9db499"> +<div id="outline-container-org4fd2e61" class="outline-4"> +<h4 id="org4fd2e61">Intersection:</h4> +<div class="outline-text-4" id="text-org4fd2e61"> <p> E ∩ F = {x / x ∈ E AND x ∈ F} ; x ∈ E ∩ F ⇔ x ∈ F AND x ∈ F<br /> </p> @@ -1272,9 +1301,9 @@ x ∉ E ∩ F ⇔ x ∉ E OR x ∉ F<br /> </p> </div> </div> -<div id="outline-container-org0946751" class="outline-4"> -<h4 id="org0946751">Union:</h4> -<div class="outline-text-4" id="text-org0946751"> +<div id="outline-container-orgabd991c" class="outline-4"> +<h4 id="orgabd991c">Union:</h4> +<div class="outline-text-4" id="text-orgabd991c"> <p> E ∪ F = {x / x ∈ E OR x ∈ F} ; x ∈ E ∪ F ⇔ x ∈ F OR x ∈ F<br /> </p> @@ -1285,17 +1314,17 @@ x ∉ E ∪ F ⇔ x ∉ E AND x ∉ F<br /> </p> </div> </div> -<div id="outline-container-org8fb5e39" class="outline-4"> -<h4 id="org8fb5e39">Difference between two sets:</h4> -<div class="outline-text-4" id="text-org8fb5e39"> +<div id="outline-container-org083102e" class="outline-4"> +<h4 id="org083102e">Difference between two sets:</h4> +<div class="outline-text-4" id="text-org083102e"> <p> E(Which is also written as : E - F) = {x / x ∈ E and x ∉ F}<br /> </p> </div> </div> -<div id="outline-container-org17ec98b" class="outline-4"> -<h4 id="org17ec98b">Complimentary set:</h4> -<div class="outline-text-4" id="text-org17ec98b"> +<div id="outline-container-orgcc1969e" class="outline-4"> +<h4 id="orgcc1969e">Complimentary set:</h4> +<div class="outline-text-4" id="text-orgcc1969e"> <p> If F ⊂ E. E - F is the complimentary of F in E.<br /> </p> @@ -1306,52 +1335,52 @@ FCE = {x /x ∈ E AND x ∉ F} <b>ONLY WHEN F IS A SUBSET OF E</b><br /> </p> </div> </div> -<div id="outline-container-org61d7d25" class="outline-4"> -<h4 id="org61d7d25">Symentrical difference</h4> -<div class="outline-text-4" id="text-org61d7d25"> +<div id="outline-container-org72511ff" class="outline-4"> +<h4 id="org72511ff">Symmetrical difference</h4> +<div class="outline-text-4" id="text-org72511ff"> <p> E Δ F = (E - F) ∪ (F - E) ; = (E ∪ F) - (E ∩ F)<br /> </p> </div> </div> </div> -<div id="outline-container-org1e0c21a" class="outline-3"> -<h3 id="org1e0c21a">Proprieties :</h3> -<div class="outline-text-3" id="text-org1e0c21a"> +<div id="outline-container-org44b1b96" class="outline-3"> +<h3 id="org44b1b96">Proprieties :</h3> +<div class="outline-text-3" id="text-org44b1b96"> <p> Let E,F and G be 3 sets. We have :<br /> </p> </div> -<div id="outline-container-orgaed74b7" class="outline-4"> -<h4 id="orgaed74b7">Commutativity:</h4> -<div class="outline-text-4" id="text-orgaed74b7"> +<div id="outline-container-orga3eac79" class="outline-4"> +<h4 id="orga3eac79">Commutativity:</h4> +<div class="outline-text-4" id="text-orga3eac79"> <p> E ∩ F = F ∩ E<br /> E ∪ F = F ∪ E<br /> </p> </div> </div> -<div id="outline-container-org1d74822" class="outline-4"> -<h4 id="org1d74822">Associativity:</h4> -<div class="outline-text-4" id="text-org1d74822"> +<div id="outline-container-org1a9121a" class="outline-4"> +<h4 id="org1a9121a">Associativity:</h4> +<div class="outline-text-4" id="text-org1a9121a"> <p> E ∩ (F ∩ G) = (E ∩ F) ∩ G<br /> E ∪ (F ∪ G) = (E ∪ F) ∪ G<br /> </p> </div> </div> -<div id="outline-container-org667f4dd" class="outline-4"> -<h4 id="org667f4dd">Distributivity:</h4> -<div class="outline-text-4" id="text-org667f4dd"> +<div id="outline-container-orgaf7fe3b" class="outline-4"> +<h4 id="orgaf7fe3b">Distributivity:</h4> +<div class="outline-text-4" id="text-orgaf7fe3b"> <p> E ∩ (F ∪ G) = (E ∩ F) ∪ (E ∩ G)<br /> E ∪ (F ∩ G) = (E ∪ F) ∩ (E ∪ G)<br /> </p> </div> </div> -<div id="outline-container-org31c4f57" class="outline-4"> -<h4 id="org31c4f57">Lois de Morgan:</h4> -<div class="outline-text-4" id="text-org31c4f57"> +<div id="outline-container-org658b728" class="outline-4"> +<h4 id="org658b728">Lois de Morgan:</h4> +<div class="outline-text-4" id="text-org658b728"> <p> If E ⊂ G and F ⊂ G ;<br /> </p> @@ -1361,33 +1390,33 @@ If E ⊂ G and F ⊂ G ;<br /> </p> </div> </div> -<div id="outline-container-orgc30ba1d" class="outline-4"> -<h4 id="orgc30ba1d">An other one:</h4> -<div class="outline-text-4" id="text-orgc30ba1d"> +<div id="outline-container-org4f0dd58" class="outline-4"> +<h4 id="org4f0dd58">An other one:</h4> +<div class="outline-text-4" id="text-org4f0dd58"> <p> E - (F ∩ G) = (E-F) ∪ (E-G) ; E - (F ∪ G) = (E-F) ∩ (E-G)<br /> </p> </div> </div> -<div id="outline-container-orgac52c86" class="outline-4"> -<h4 id="orgac52c86">An other one:</h4> -<div class="outline-text-4" id="text-orgac52c86"> +<div id="outline-container-orgdd60033" class="outline-4"> +<h4 id="orgdd60033">An other one:</h4> +<div class="outline-text-4" id="text-orgdd60033"> <p> E ∩ ∅ = ∅ ; E ∪ ∅ = E<br /> </p> </div> </div> -<div id="outline-container-orge0de23e" class="outline-4"> -<h4 id="orge0de23e">And an other one:</h4> -<div class="outline-text-4" id="text-orge0de23e"> +<div id="outline-container-orgbf5feb1" class="outline-4"> +<h4 id="orgbf5feb1">And an other one:</h4> +<div class="outline-text-4" id="text-orgbf5feb1"> <p> E ∩ (F Δ G) = (E ∩ F) Δ (E ∩ G)<br /> </p> </div> </div> -<div id="outline-container-org8277b0b" class="outline-4"> -<h4 id="org8277b0b">And the last one:</h4> -<div class="outline-text-4" id="text-org8277b0b"> +<div id="outline-container-orgefabd47" class="outline-4"> +<h4 id="orgefabd47">And the last one:</h4> +<div class="outline-text-4" id="text-orgefabd47"> <p> E Δ ∅ = E ; E Δ E = ∅<br /> </p> @@ -1395,16 +1424,16 @@ E Δ ∅ = E ; E Δ E = ∅<br /> </div> </div> </div> -<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"> +<div id="outline-container-orgc9ed06c" class="outline-2"> +<h2 id="orgc9ed06c">5eme cours: L’ensemble des parties d’un ensemble <i>Oct 16</i></h2> +<div class="outline-text-2" id="text-orgc9ed06c"> <p> Let E be a set. We define P(E) as the set of all parts of E : <b>P(E) = {X/X ⊂ E}</b><br /> </p> </div> -<div id="outline-container-org2c7d514" class="outline-4"> -<h4 id="org2c7d514">Notes :</h4> -<div class="outline-text-4" id="text-org2c7d514"> +<div id="outline-container-org7a1da3c" class="outline-4"> +<h4 id="org7a1da3c">Notes :</h4> +<div class="outline-text-4" id="text-org7a1da3c"> <p> ∅ ∈ P(E) ; E ∈ P(E)<br /> </p> @@ -1415,17 +1444,17 @@ cardinal E = n <i>The number of terms in E</i> , cardinal P(E) = 2^n <i>The numb </p> </div> </div> -<div id="outline-container-org76da8f4" class="outline-4"> -<h4 id="org76da8f4">Examples :</h4> -<div class="outline-text-4" id="text-org76da8f4"> +<div id="outline-container-orga6e5f8a" class="outline-4"> +<h4 id="orga6e5f8a">Examples :</h4> +<div class="outline-text-4" id="text-orga6e5f8a"> <p> E = {a,b,c} ; P(E)={∅, {a}, {b}, {c}, {a,b}, {b,c}, {a,c}, {a,b,c}}<br /> </p> </div> </div> -<div id="outline-container-org3da3de1" class="outline-3"> -<h3 id="org3da3de1">Partition of a set :</h3> -<div class="outline-text-3" id="text-org3da3de1"> +<div id="outline-container-org7286ec5" class="outline-3"> +<h3 id="org7286ec5">Partition of a set :</h3> +<div class="outline-text-3" id="text-org7286ec5"> <p> We say that <b>A</b> is a partition of E if:<br /> </p> @@ -1436,16 +1465,16 @@ We say that <b>A</b> is a partition of E if:<br /> </ol> </div> </div> -<div id="outline-container-org077b994" class="outline-3"> -<h3 id="org077b994">Cartesian products :</h3> -<div class="outline-text-3" id="text-org077b994"> +<div id="outline-container-orgd8e00ac" class="outline-3"> +<h3 id="orgd8e00ac">Cartesian products :</h3> +<div class="outline-text-3" id="text-orgd8e00ac"> <p> Let E and F be two sets, the set EXF = {(x,y)/ x ∈ E AND y ∈ F} is called the Cartesian product of E and F<br /> </p> </div> -<div id="outline-container-org72822f5" class="outline-4"> -<h4 id="org72822f5">Example :</h4> -<div class="outline-text-4" id="text-org72822f5"> +<div id="outline-container-orgdb491ee" class="outline-4"> +<h4 id="orgdb491ee">Example :</h4> +<div class="outline-text-4" id="text-orgdb491ee"> <p> A = {4,5} ; B= {4,5,6} ; AxB = {(4,4), (4,5), (4,6), (5,4), (5,5), (5,6)}<br /> </p> @@ -1456,9 +1485,9 @@ BxA = {(4,4), (4,5), (5,4), (5,5), (6,4), (6,5)} ; Therefore AxB ≠ BxA<br /> </p> </div> </div> -<div id="outline-container-org04e1be3" class="outline-4"> -<h4 id="org04e1be3">Some proprieties:</h4> -<div class="outline-text-4" id="text-org04e1be3"> +<div id="outline-container-org28c23b2" class="outline-4"> +<h4 id="org28c23b2">Some proprieties:</h4> +<div class="outline-text-4" id="text-org28c23b2"> <ol class="org-ol"> <li>ExF = ∅ ⇔ E=∅ OR F=∅<br /></li> <li>ExF = FxE ⇔ E=F OR E=∅ OR F=∅<br /></li> @@ -1471,21 +1500,21 @@ BxA = {(4,4), (4,5), (5,4), (5,5), (6,4), (6,5)} ; Therefore AxB ≠ BxA<br /> </div> </div> </div> -<div id="outline-container-org416c1e1" class="outline-2"> -<h2 id="org416c1e1">Binary relations in a set :</h2> -<div class="outline-text-2" id="text-org416c1e1"> +<div id="outline-container-orgfdbe4f3" class="outline-2"> +<h2 id="orgfdbe4f3">Binary relations in a set :</h2> +<div class="outline-text-2" id="text-orgfdbe4f3"> </div> -<div id="outline-container-org5ea795f" class="outline-3"> -<h3 id="org5ea795f">Definition :</h3> -<div class="outline-text-3" id="text-org5ea795f"> +<div id="outline-container-org3696656" class="outline-3"> +<h3 id="org3696656">Definition :</h3> +<div class="outline-text-3" id="text-org3696656"> <p> Let E be a set and x,y ∈ E. If there exists a link between x and y, we say that they are tied by a relation <b>R</b> and we write <b>xRy</b><br /> </p> </div> </div> -<div id="outline-container-orgb9d678f" class="outline-3"> -<h3 id="orgb9d678f">Proprieties :</h3> -<div class="outline-text-3" id="text-orgb9d678f"> +<div id="outline-container-orgd32f673" class="outline-3"> +<h3 id="orgd32f673">Proprieties :</h3> +<div class="outline-text-3" id="text-orgd32f673"> <p> Let E be a set and R a relation defined in E<br /> </p> @@ -1497,16 +1526,16 @@ Let E be a set and R a relation defined in E<br /> </ol> </div> </div> -<div id="outline-container-org6df8952" class="outline-3"> -<h3 id="org6df8952">Equivalence relationship :</h3> -<div class="outline-text-3" id="text-org6df8952"> +<div id="outline-container-org22f460a" class="outline-3"> +<h3 id="org22f460a">Equivalence relationship :</h3> +<div class="outline-text-3" id="text-org22f460a"> <p> We say that R is a relation of equivalence in E if its reflexive, symetrical and transitive<br /> </p> </div> -<div id="outline-container-orgfbd9232" class="outline-4"> -<h4 id="orgfbd9232">Equivalence class :</h4> -<div class="outline-text-4" id="text-orgfbd9232"> +<div id="outline-container-org68ddde2" class="outline-4"> +<h4 id="org68ddde2">Equivalence class :</h4> +<div class="outline-text-4" id="text-org68ddde2"> <p> Let R be a relation of equivalence in E and a ∈ E, we call equivalence class of <b>a</b>, and we write ̅a or ȧ, or cl a the following set :<br /> </p> @@ -1517,8 +1546,8 @@ Let R be a relation of equivalence in E and a ∈ E, we call equivalence class o </p> </div> <ul class="org-ul"> -<li><a id="org1572848"></a>The quotient set :<br /> -<div class="outline-text-5" id="text-org1572848"> +<li><a id="org431774d"></a>The quotient set :<br /> +<div class="outline-text-5" id="text-org431774d"> <p> E/R = {̅a , a ∈ E}<br /> </p> @@ -1527,9 +1556,9 @@ E/R = {̅a , a ∈ E}<br /> </ul> </div> </div> -<div id="outline-container-org9a36dc1" class="outline-3"> -<h3 id="org9a36dc1">Order relationship :</h3> -<div class="outline-text-3" id="text-org9a36dc1"> +<div id="outline-container-orge976c7e" class="outline-3"> +<h3 id="orge976c7e">Order relationship :</h3> +<div class="outline-text-3" id="text-orge976c7e"> <p> Let E be a set and R be a relation defined in E. We say that R is a relation of order if its reflexive, anti-symetrical and transitive.<br /> </p> @@ -1538,9 +1567,9 @@ Let E be a set and R be a relation defined in E. We say that R is a relation of <li>The order R is called partial if ∃ x,y ∈ E xR̅y AND yR̅x<br /></li> </ol> </div> -<div id="outline-container-orgb496cba" class="outline-4"> -<h4 id="orgb496cba"><span class="todo TODO">TODO</span> Examples :</h4> -<div class="outline-text-4" id="text-orgb496cba"> +<div id="outline-container-org1f19847" class="outline-4"> +<h4 id="org1f19847"><span class="todo TODO">TODO</span> Examples :</h4> +<div class="outline-text-4" id="text-org1f19847"> <p> ∀x,y ∈ ℝ , xRy ⇔ x²-y²=x-y<br /> </p> @@ -1552,17 +1581,17 @@ Let E be a set and R be a relation defined in E. We say that R is a relation of </div> </div> </div> -<div id="outline-container-org54d5489" class="outline-2"> -<h2 id="org54d5489">TP exercices <i>Oct 20</i> :</h2> -<div class="outline-text-2" id="text-org54d5489"> +<div id="outline-container-orgc956a73" class="outline-2"> +<h2 id="orgc956a73">TP exercices <i>Oct 20</i> :</h2> +<div class="outline-text-2" id="text-orgc956a73"> </div> -<div id="outline-container-orgdfd55ca" class="outline-3"> -<h3 id="orgdfd55ca">Exercice 3 :</h3> -<div class="outline-text-3" id="text-orgdfd55ca"> +<div id="outline-container-org15b0e75" class="outline-3"> +<h3 id="org15b0e75">Exercice 3 :</h3> +<div class="outline-text-3" id="text-org15b0e75"> </div> -<div id="outline-container-org4100fe3" class="outline-4"> -<h4 id="org4100fe3">Question 3</h4> -<div class="outline-text-4" id="text-org4100fe3"> +<div id="outline-container-orgb132892" class="outline-4"> +<h4 id="orgb132892">Question 3</h4> +<div class="outline-text-4" id="text-orgb132892"> <p> Montrer par l’absurde que P : ∀x ∈ ℝ*, √(4+x³) ≠ 2 + x³/4 est vraies<br /> </p> @@ -1579,13 +1608,13 @@ x = 0 . Or, x appartiens a ℝ\{0}, donc P̅ est fausse. Ce qui est equivalent a </div> </div> </div> -<div id="outline-container-org019b5e0" class="outline-3"> -<h3 id="org019b5e0">Exercice 4 :</h3> -<div class="outline-text-3" id="text-org019b5e0"> +<div id="outline-container-org9a4006b" class="outline-3"> +<h3 id="org9a4006b">Exercice 4 :</h3> +<div class="outline-text-3" id="text-org9a4006b"> </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"> +<div id="outline-container-org43cf6d6" class="outline-4"> +<h4 id="org43cf6d6"><span class="done DONE">DONE</span> Question 1 :</h4> +<div class="outline-text-4" id="text-org43cf6d6"> <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 /> @@ -1613,10 +1642,228 @@ De (a) et (b) on conclus que la proposition de départ est vraie<br /> </div> </div> </div> +<div id="outline-container-org429ab91" class="outline-2"> +<h2 id="org429ab91">Chapter 3 : Applications</h2> +<div class="outline-text-2" id="text-org429ab91"> +</div> +<div id="outline-container-org4a4b3cf" class="outline-3"> +<h3 id="org4a4b3cf">3.1 Generalities about applications :</h3> +<div class="outline-text-3" id="text-org4a4b3cf"> +</div> +<div id="outline-container-org0cac0c6" class="outline-4"> +<h4 id="org0cac0c6">Definition :</h4> +<div class="outline-text-4" id="text-org0cac0c6"> +<p> +Let E and F be two sets.<br /> +</p> +<ol class="org-ol"> +<li>We call a function of the set E to the set F any relation from E to F such as for any element of E, we can find <span class="underline">at most one</span> element of F that corresponds to it.<br /></li> +<li>We call an application of the set E to the set F a relation from E to F such as for any element of E, we can find <span class="underline">one and only one</span> element of F that corresponds to it.<br /></li> +<li><p> +f: E<sub>1</sub> —> F<sub>1</sub> ; g: E<sub>2</sub> —> F<sub>2</sub> ; f ≡ g ⇔ [E<sub>1 </sub>= E<sub>2</sub> ; F<sub>1</sub> = F<sub>2</sub> ; f(x) = g(x) ∀x ∈ E<sub>1</sub><br /> +</p> + +<p> +Generally speaking, we schematize a function or an application by this writing :<br /> +</p> +<p class="verse"> +f : E —> F<br /> +    x —> f(x)=y<br /> +   Γ = {(x , f(x))/ x ∈ E ; f(x) ∈ F} is the graph of f<br /> +</p></li> +</ol> +</div> +<ul class="org-ul"> +<li><a id="org1570343"></a>Some examples :<br /> +<ul class="org-ul"> +<li><a id="orgdafc0e8"></a>Ex1:<br /> +<div class="outline-text-6" id="text-orgdafc0e8"> +<p class="verse"> +f : ℝ —> ℝ<br /> +    x —> f(x) = (x-1)/x<br /> +is a function, because 0 does NOT have a corresponding element using that relation.<br /> +</p> +</div> +</li> +<li><a id="orgcc5e730"></a>Ex2:<br /> +<div class="outline-text-6" id="text-orgcc5e730"> +<p class="verse"> +f : ℝ<sup>*</sup> —> ℝ<br /> +    x —> f(x)= (x-1)/x<br /> +is, however, an application<br /> +</p> +</div> +</li> +</ul> +</li> +</ul> +</div> +<div id="outline-container-orgc048e93" class="outline-4"> +<h4 id="orgc048e93">Restriction and prolongation of an application :</h4> +<div class="outline-text-4" id="text-orgc048e93"> +<p> +Let f : E -> F an application and E<sub>1</sub> ⊂ E therefore :<br /> +</p> +<p class="verse"> +g : E<sub>1</sub> -> F<br /> +g(x) = f(x) ∀x ∈ E<sub>1</sub><br /> +<br /> +g is called the <b>restriction</b> of f to E<sub>1</sub>. And f is called the <b>prolongation</b> of g to E.<br /> +</p> +</div> +<ul class="org-ul"> +<li><a id="org771da73"></a>Example<br /> +<div class="outline-text-5" id="text-org771da73"> +<p class="verse"> +f : ℝ —> ℝ<br /> +    x —> f(x) = x<sup>2</sup><br /> +<br /> +g : [0 , <del>∞[ —> ℝ<br /> +    x —> g(x) = x²<br /> +<br /> +g is called the <b>restriction</b> of f to ℝ^{</del>}. And f is called the <b>prolongation</b> of g to ℝ.<br /> +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-org8e61361" class="outline-4"> +<h4 id="org8e61361">Composition of applications :</h4> +<div class="outline-text-4" id="text-org8e61361"> +<p> +Let E,F, and G be three sets, f: E -> F and g: F -> G are two applications. We define their composition, symbolized by g<sub>o</sub>f as follow :<br /> +</p> + + +<p> +g<sub>o</sub>f : E -> G . ∀x ∈ E (g<sub>o</sub>f)<sub>(x)</sub>= g(f(x))<br /> +</p> +</div> +</div> +</div> +<div id="outline-container-org5c096db" class="outline-3"> +<h3 id="org5c096db">3.2 Injection, surjection and bijection :</h3> +<div class="outline-text-3" id="text-org5c096db"> +<p> +Let f: E -> F be an application :<br /> +</p> +<ol class="org-ol"> +<li>We say that f is injective if : ∀x,x’ ∈ E : f(x) = f(x’) ⇒ x = x’<br /></li> +<li>We say that f is surjective if : ∀ y ∈ F , ∃ x ∈ E : y = f(x)<br /></li> +<li>We say that if is bijective if it’s both injective and surjective at the same time.<br /></li> +</ol> +</div> +<div id="outline-container-org4162b56" class="outline-4"> +<h4 id="org4162b56">Proposition :</h4> +<div class="outline-text-4" id="text-org4162b56"> +<p> +Let f : E -> F be an application. Therefore:<br /> +</p> +<ol class="org-ol"> +<li>f is injective ⇔ y = f(x) has at most one solution.<br /></li> +<li>f is surjective ⇔ y = f(x) has at least one solution.<br /></li> +<li>f is bijective ⇔ y = f(x) has a single and unique solution.<br /></li> +</ol> +</div> +</div> +</div> +<div id="outline-container-org736de6c" class="outline-3"> +<h3 id="org736de6c">3.3 Reciprocal applications :</h3> +<div class="outline-text-3" id="text-org736de6c"> +</div> +<div id="outline-container-orgafb7f85" class="outline-4"> +<h4 id="orgafb7f85">Def :</h4> +<div class="outline-text-4" id="text-orgafb7f85"> +<p> +Let f : E -> F a bijective application. So there exists an application named f<sup>-1</sup> : F -> E such as : y = f(x) ⇔ x = f<sup>-1</sup>(y)<br /> +</p> +</div> +</div> +<div id="outline-container-orgec3a3d6" class="outline-4"> +<h4 id="orgec3a3d6">Theorem :</h4> +<div class="outline-text-4" id="text-orgec3a3d6"> +<p> +Let f : E -> F be a bijective application. Therefore its reciprocal f<sup>-1</sup> verifies : f<sup>-1</sup><sub>o</sub>f=Id<sub>E </sub>; f<sub>o</sub>f<sup>-1</sup>=Id<sub>F</sub> Or :<br /> +</p> + + +<p> +Id<sub>E</sub> : E -> E ; x -> Id<sub>E</sub>(x) = x<br /> +</p> +</div> +</div> +<div id="outline-container-org940e2d6" class="outline-4"> +<h4 id="org940e2d6">Some proprieties :</h4> +<div class="outline-text-4" id="text-org940e2d6"> +<ol class="org-ol"> +<li>(f<sup>-1</sup>)<sup>-1</sup> = f<br /></li> +<li>(g<sub>o</sub>f)⁻¹ = f⁻¹<sub>o</sub>g⁻¹<br /></li> +<li>The graphs of f and f⁻¹ are symmetrical to each other by the first bis-sectrice of the equation y = x<br /></li> +</ol> +</div> +</div> +</div> +<div id="outline-container-org2d173c2" class="outline-3"> +<h3 id="org2d173c2">3.4 Direct Image and reciprocal Image :</h3> +<div class="outline-text-3" id="text-org2d173c2"> +</div> +<div id="outline-container-org769c809" class="outline-4"> +<h4 id="org769c809">Direct Image :</h4> +<div class="outline-text-4" id="text-org769c809"> +<p> +Let f: E-> F be an application and A ⊂ E. We call a direct image of A by f, and we symbolize as f(A) the subset of F defined by :<br /> +</p> + + +<p> +f(A) = {f(x)/ x ∈ A} ; = { y ∈ F ∃ x ∈ A y=f(x)}<br /> +</p> +</div> +<ul class="org-ul"> +<li><a id="org492a9ac"></a>Example :<br /> +<div class="outline-text-5" id="text-org492a9ac"> +<p class="verse"> +f: ℝ -> ℝ<br /> +   x -> f(x) = x²<br /> +A = {0,4}<br /> +f(A) = {f(0), f(4)} = {0, 16}<br /> +</p> +</div> +</li> +</ul> +</div> +<div id="outline-container-org7d705d3" class="outline-4"> +<h4 id="org7d705d3">Reciprocal image :</h4> +<div class="outline-text-4" id="text-org7d705d3"> +<p> +Let f: E -> F be an application and B ⊂ F. We call the reciprocal image of E by F the subset f<sup>-1</sup>(B) :<br /> +</p> + + +<p> +f<sup>-1</sup>(B) = {x ∈ E/f(x) ∈ B} ; x ∈ f<sup>-1</sup>(B) ⇔ f(x) ∈ B<br /> +</p> +</div> +<ul class="org-ul"> +<li><a id="org686c1dc"></a>Example :<br /> +<div class="outline-text-5" id="text-org686c1dc"> +<p class="verse"> +f: ℝ -> ℝ<br /> +   x -> f(x) = x²<br /> +B = {1,9,4}<br /> +f<sup>-1</sup>(B) = {1,-1,2,-2,3,-3}<br /> +      = {x ∈ ℝ/x² ∈ {1,4,9}}<br /> +</p> +</div> +</li> +</ul> +</div> +</div> +</div> </div> <div id="postamble" class="status"> <p class="author">Author: Crystal</p> -<p class="date">Created: 2023-10-20 Fri 15:12</p> +<p class="date">Created: 2023-10-23 Mon 19:39</p> </div> </body> </html> \ No newline at end of file |