Finally, an update

author: Crystal <crystal@wizard.tower> 2023-10-17 22:42:30 +0100
committer: Crystal <crystal@wizard.tower> 2023-10-17 22:42:30 +0100
commit: 16e5f6e90b603bc62d4c0af24d79ca5e6121ac6a (patch)
tree: b8ea1a57004ec9498fd41988b84f355deb2fff78
parent: 7315416eaee4c833bff711de48975ed95bedf3fd (diff)
download: www-16e5f6e90b603bc62d4c0af24d79ca5e6121ac6a.tar.gz
2 files changed, 389 insertions, 174 deletions
diff --git a/src/org/uni_notes/algebra1.org b/src/org/uni_notes/algebra1.org
index 1126423..21e41ef 100755
--- a/src/org/uni_notes/algebra1.org
+++ b/src/org/uni_notes/algebra1.org
@@ -494,3 +494,61 @@ E ∩ ∅ = ∅ ; E ∪ ∅ = E
 E ∩ (F Δ G) = (E ∩ F) Δ (E ∩ G)
 *** And the last one:
 E Δ ∅ = E ; E Δ E = ∅
+* 5eme cours: L'ensemble des parties d'un ensemble /Oct 16/
+Let E be a set. We define P(E) as the set of all parts of E : *P(E) = {X/X ⊂ E}*
+
+
+*** Notes :
+∅ ∈ P(E) ; E ∈ P(E)
+
+
+cardinal E = n /The number of terms in E/ , cardinal P(E) = 2^n /The number of all parts of E/
+
+*** Examples :
+E = {a,b,c} // P(E)={∅, {a}, {b}, {c}, {a,b}, {b,c}, {a,c}, {a,b,c}}
+
+** Partition of a set :
+We say that *A* is a partition of E if:
+a. ∀ x ∈ A , x ≠ 0
+b. All the elements of *A* are two by two disjoint. Or in other terms, there should not be two elements that intersects with each other.
+c. The reunion of all elements of *A* is equal to E
+** Cartesian products :
+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
+*** Example :
+A = {4,5} ; B= {4,5,6} // AxB = {(4,4), (4,5), (4,6), (5,4), (5,5), (5,6)}
+
+
+BxA = {(4,4), (4,5), (5,4), (5,5), (6,4), (6,5)} // Therefore AxB ≠ BxA
+*** Some proprieties:
+1. ExF = ∅ ⇔ E=∅ OR F=∅
+2. ExF = FxE ⇔ E=F OR E=∅ OR F=∅
+3. E x (F∪G) = (ExF) ∪ (ExG)
+4. (E∪F) x G = (ExG) ∪ (FxG)
+5. (E∪F) ∩ (GxH) = (E ∩ G) x (F ∩ H)
+6. Generally speaking : (ExF) ∪ (GxH) ≠ (E∪G) x (F∪H)
+* Binary relations in a set :
+** Definition :
+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 *R* and we write *xRy*
+** Proprieties :
+Let E be a set and R a relation defined in E
+1. We say that R is reflexive if ∀ x ∈ E, xRx (for any element x in E,x is related to itself)
+2. We say that R is symmetrical if ∀ x,y ∈ E , xRy ⇒ yRx
+3. We say that R is transitive if ∀ x,y,z ∈ E (xRy , yRz) ⇒ xRz
+4. We say that R is anti-symmetrical if ∀ x,y ∈ E xRy AND yRx ⇒ x = y
+** Equivalence relationship :
+We say that R is a relation of equivalence in E if its reflexive, symetrical and transitive
+*** Equivalence class :
+Let R be a relation of equivalence in E and a ∈ E, we call equivalence class of *a*, and we write ̅a or ȧ, or cl a the following set :
+
+
+*a̅ = {y ∈ E/ y R a}*
+**** The quotient set :
+E/R = {̅a , a ∈ E}
+** Order relationship :
+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.
+1. The order R is called total if ∀ x,y ∈ E xRy OR yRx
+2. The order R is called partial if ∃ x,y ∈ E xR̅y AND yR̅x
+*** TODO Examples :
+∀x,y ∈ ℝ , xRy ⇔ x²-y²=x-y
+1. Prove that R is an equivalence relation
+2. Let a ∈ ℝ, find ̅a
diff --git a/uni_notes/algebra.html b/uni_notes/algebra.html
index 9323119..2129e72 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>
-<!-- 2023-10-13 Fri 16:58 -->
+<!-- 2023-10-17 Tue 22:32 -->
 <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>
@@ -47,13 +47,13 @@
 <body>
 <div id="content" class="content">
 <h1 class="title">Algebra 1</h1>
-<div id="outline-container-orgc134a5b" class="outline-2">
-<h2 id="orgc134a5b">Contenu de la Matiére</h2>
-<div class="outline-text-2" id="text-orgc134a5b">
+<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>
-<div id="outline-container-orgae00938" class="outline-3">
-<h3 id="orgae00938">Rappels et compléments (11H)</h3>
-<div class="outline-text-3" id="text-orgae00938">
+<div id="outline-container-org2a89be2" class="outline-3">
+<h3 id="org2a89be2">Rappels et compléments (11H)</h3>
+<div class="outline-text-3" id="text-org2a89be2">
 <ul class="org-ul">
 <li>Logique mathématique et méthodes du raisonnement mathématique</li>
 <li>Ensembles et Relations</li>
@@ -61,9 +61,9 @@
 </ul>
 </div>
 </div>
-<div id="outline-container-org0eb35c9" class="outline-3">
-<h3 id="org0eb35c9">Structures Algébriques (11H)</h3>
-<div class="outline-text-3" id="text-org0eb35c9">
+<div id="outline-container-orgfcd7f3f" class="outline-3">
+<h3 id="orgfcd7f3f">Structures Algébriques (11H)</h3>
+<div class="outline-text-3" id="text-orgfcd7f3f">
 <ul class="org-ul">
 <li>Groupes et morphisme de groupes</li>
 <li>Anneaux et morphisme d&rsquo;anneaux</li>
@@ -71,9 +71,9 @@
 </ul>
 </div>
 </div>
-<div id="outline-container-org4a088f6" class="outline-3">
-<h3 id="org4a088f6">Polynômes et fractions rationnelles</h3>
-<div class="outline-text-3" id="text-org4a088f6">
+<div id="outline-container-org03cbf05" class="outline-3">
+<h3 id="org03cbf05">Polynômes et fractions rationnelles</h3>
+<div class="outline-text-3" id="text-org03cbf05">
 <ul class="org-ul">
 <li>Notion du polynôme à une indéterminée á coefficients dans un anneau</li>
 <li>Opérations Algébriques sur les polynômes</li>
@@ -86,9 +86,9 @@
 </div>
 </div>
 </div>
-<div id="outline-container-org73264c6" class="outline-2">
-<h2 id="org73264c6">Premier cours : Logique mathématique et méthodes du raisonnement mathématique <i>Sep 25</i> :</h2>
-<div class="outline-text-2" id="text-org73264c6">
+<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">
 <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&rsquo;s also give the value <b>1</b> to each <b>True</b> proposition and <b>0</b> to each false one.
 </p>
@@ -438,13 +438,13 @@ 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>
 </p>
 </div>
-<div id="outline-container-org74d9557" class="outline-3">
-<h3 id="org74d9557">Properties:</h3>
-<div class="outline-text-3" id="text-org74d9557">
+<div id="outline-container-org5604636" class="outline-3">
+<h3 id="org5604636">Properties:</h3>
+<div class="outline-text-3" id="text-org5604636">
 </div>
-<div id="outline-container-orgc7f1d03" class="outline-4">
-<h4 id="orgc7f1d03"><b>Absorption</b>:</h4>
-<div class="outline-text-4" id="text-orgc7f1d03">
+<div id="outline-container-orgfffc23d" class="outline-4">
+<h4 id="orgfffc23d"><b>Absorption</b>:</h4>
+<div class="outline-text-4" id="text-orgfffc23d">
 <p>
 (P ∨ P) ⇔ P
 </p>
@@ -454,9 +454,9 @@ A proposition is equivalent to another only when both of them have <b>the same v
 </p>
 </div>
 </div>
-<div id="outline-container-orgcb729de" class="outline-4">
-<h4 id="orgcb729de"><b>Commutativity</b>:</h4>
-<div class="outline-text-4" id="text-orgcb729de">
+<div id="outline-container-orgd43aeb7" class="outline-4">
+<h4 id="orgd43aeb7"><b>Commutativity</b>:</h4>
+<div class="outline-text-4" id="text-orgd43aeb7">
 <p>
 (P ∧ Q) ⇔ (Q ∧ P)
 </p>
@@ -466,9 +466,9 @@ A proposition is equivalent to another only when both of them have <b>the same v
 </p>
 </div>
 </div>
-<div id="outline-container-org4ae8933" class="outline-4">
-<h4 id="org4ae8933"><b>Associativity</b>:</h4>
-<div class="outline-text-4" id="text-org4ae8933">
+<div id="outline-container-org9e5868e" class="outline-4">
+<h4 id="org9e5868e"><b>Associativity</b>:</h4>
+<div class="outline-text-4" id="text-org9e5868e">
 <p>
 P ∧ (Q ∧ R) ⇔ (P ∧ Q) ∧ R
 </p>
@@ -478,9 +478,9 @@ P ∨ (Q ∨ R) ⇔ (P ∨ Q) ∨ R
 </p>
 </div>
 </div>
-<div id="outline-container-org095f4a6" class="outline-4">
-<h4 id="org095f4a6"><b>Distributivity</b>:</h4>
-<div class="outline-text-4" id="text-org095f4a6">
+<div id="outline-container-orga530d13" class="outline-4">
+<h4 id="orga530d13"><b>Distributivity</b>:</h4>
+<div class="outline-text-4" id="text-orga530d13">
 <p>
 P ∧ (Q ∨ R) ⇔ (P ∧ Q) ∨ (P ∧ R)
 </p>
@@ -490,9 +490,9 @@ P ∨ (Q ∧ R) ⇔ (P ∨ Q) ∧ (P ∨ R)
 </p>
 </div>
 </div>
-<div id="outline-container-orgf7e29ee" class="outline-4">
-<h4 id="orgf7e29ee"><b>Neutral element</b>:</h4>
-<div class="outline-text-4" id="text-orgf7e29ee">
+<div id="outline-container-org7d55048" class="outline-4">
+<h4 id="org7d55048"><b>Neutral element</b>:</h4>
+<div class="outline-text-4" id="text-org7d55048">
 <p>
 <i>We define proposition <b>T</b> to be always <b>true</b> and <b>F</b> to be always <b>false</b></i>
 </p>
@@ -506,9 +506,9 @@ P ∨ F ⇔ P
 </p>
 </div>
 </div>
-<div id="outline-container-org9bfee59" class="outline-4">
-<h4 id="org9bfee59"><b>Negation of a conjunction &amp; a disjunction</b>:</h4>
-<div class="outline-text-4" id="text-org9bfee59">
+<div id="outline-container-org7422610" class="outline-4">
+<h4 id="org7422610"><b>Negation of a conjunction &amp; a disjunction</b>:</h4>
+<div class="outline-text-4" id="text-org7422610">
 <p>
 Now we won&rsquo;t use bars here because my lazy ass doesn&rsquo;t know how, so instead I will use not()!!!
 </p>
@@ -526,25 +526,25 @@ not(<b>P ∨ Q</b>) ⇔ P̅ ∧ Q̅
 </p>
 </div>
 </div>
-<div id="outline-container-orgd144fb3" class="outline-4">
-<h4 id="orgd144fb3"><b>Transitivity</b>:</h4>
-<div class="outline-text-4" id="text-orgd144fb3">
+<div id="outline-container-org4145760" class="outline-4">
+<h4 id="org4145760"><b>Transitivity</b>:</h4>
+<div class="outline-text-4" id="text-org4145760">
 <p>
 [(P ⇒ Q) AND (Q ⇒ R)] ⇔ P ⇒ R
 </p>
 </div>
 </div>
-<div id="outline-container-orgbc26f01" class="outline-4">
-<h4 id="orgbc26f01"><b>Contraposition</b>:</h4>
-<div class="outline-text-4" id="text-orgbc26f01">
+<div id="outline-container-org245af1d" class="outline-4">
+<h4 id="org245af1d"><b>Contraposition</b>:</h4>
+<div class="outline-text-4" id="text-org245af1d">
 <p>
 (P ⇒ Q) ⇔ (Q̅ ⇒ P̅)
 </p>
 </div>
 </div>
-<div id="outline-container-org70fd37a" class="outline-4">
-<h4 id="org70fd37a">God only knows what this property is called:</h4>
-<div class="outline-text-4" id="text-org70fd37a">
+<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">
 <p>
 <i>If</i>
 </p>
@@ -571,17 +571,17 @@ Q is always true
 </div>
 </div>
 </div>
-<div id="outline-container-org6dd9c74" class="outline-3">
-<h3 id="org6dd9c74">Some exercices I found online :</h3>
-<div class="outline-text-3" id="text-org6dd9c74">
+<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>
-<div id="outline-container-org5a46794" class="outline-4">
-<h4 id="org5a46794">USTHB 2022/2023 Section B :</h4>
-<div class="outline-text-4" id="text-org5a46794">
+<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>
 <ul class="org-ul">
-<li><a id="orgdcdfa08"></a>Exercice 1: Démontrer les équivalences suivantes:<br />
-<div class="outline-text-5" id="text-orgdcdfa08">
+<li><a id="orgd7d6ce9"></a>Exercice 1: Démontrer les équivalences suivantes:<br />
+<div class="outline-text-5" id="text-orgd7d6ce9">
 <ol class="org-ol">
 <li><p>
 (P ⇒ Q) ⇔ (Q̅ ⇒ P̅)
@@ -635,8 +635,8 @@ Literally the same as above 🩷
 </ol>
 </div>
 </li>
-<li><a id="orgfc2dd28"></a>Exercice 2: Dire si les propositions suivantes sont vraies ou fausses, et les nier:<br />
-<div class="outline-text-5" id="text-orgfc2dd28">
+<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">
 <ol class="org-ol">
 <li><p>
 ∀x ∈ ℝ ,∃y ∈ ℝ*+, tels que e^x = y
@@ -769,13 +769,13 @@ y + x &lt; 8
 </div>
 </div>
 </div>
-<div id="outline-container-org8e69635" class="outline-2">
-<h2 id="org8e69635">2éme cours <i>Oct 2</i></h2>
-<div class="outline-text-2" id="text-org8e69635">
+<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>
-<div id="outline-container-org1d4ffa3" class="outline-3">
-<h3 id="org1d4ffa3">Quantifiers</h3>
-<div class="outline-text-3" id="text-org1d4ffa3">
+<div id="outline-container-org22b148b" class="outline-3">
+<h3 id="org22b148b">Quantifiers</h3>
+<div class="outline-text-3" id="text-org22b148b">
 <p>
 A propriety P can depend on a parameter x
 </p>
@@ -791,8 +791,8 @@ A propriety P can depend on a parameter x
 </p>
 </div>
 <ul class="org-ul">
-<li><a id="orgd0b7f53"></a>Example<br />
-<div class="outline-text-6" id="text-orgd0b7f53">
+<li><a id="org4afa1df"></a>Example<br />
+<div class="outline-text-6" id="text-org4afa1df">
 <p>
 P(x) : x+1≥0
 </p>
@@ -803,13 +803,13 @@ P(X) is True or False depending on the values of x
 </div>
 </li>
 </ul>
-<div id="outline-container-orga7cd185" class="outline-4">
-<h4 id="orga7cd185">Proprieties</h4>
-<div class="outline-text-4" id="text-orga7cd185">
+<div id="outline-container-org8b437f3" class="outline-4">
+<h4 id="org8b437f3">Proprieties</h4>
+<div class="outline-text-4" id="text-org8b437f3">
 </div>
 <ul class="org-ul">
-<li><a id="org7460082"></a>Propriety Number 1:<br />
-<div class="outline-text-5" id="text-org7460082">
+<li><a id="org6d0c06f"></a>Propriety Number 1:<br />
+<div class="outline-text-5" id="text-org6d0c06f">
 <p>
 The negation of the universal quantifier is the existential quantifier, and vice-versa :
 </p>
@@ -820,8 +820,8 @@ The negation of the universal quantifier is the existential quantifier, and vice
 </ul>
 </div>
 <ul class="org-ul">
-<li><a id="org27b8375"></a>Example:<br />
-<div class="outline-text-6" id="text-org27b8375">
+<li><a id="orgd242e81"></a>Example:<br />
+<div class="outline-text-6" id="text-orgd242e81">
 <p>
 ∀ x ≥ 1  x² &gt; 5 ⇔ ∃ x ≥ 1 x² &lt; 5
 </p>
@@ -829,8 +829,8 @@ The negation of the universal quantifier is the existential quantifier, and vice
 </li>
 </ul>
 </li>
-<li><a id="org21aa647"></a>Propriety Number 2:<br />
-<div class="outline-text-5" id="text-org21aa647">
+<li><a id="orgd7c2c1d"></a>Propriety Number 2:<br />
+<div class="outline-text-5" id="text-orgd7c2c1d">
 <p>
 <b>∀x ∈ E, [P(x) ∧ Q(x)] ⇔ [∀ x ∈ E, P(x)] ∧ [∀ x ∈ E, Q(x)]</b>
 </p>
@@ -841,8 +841,8 @@ The propriety &ldquo;For any value of x from a set E , P(x) and Q(x)&rdquo; is e
 </p>
 </div>
 <ul class="org-ul">
-<li><a id="orgb4e2845"></a>Example :<br />
-<div class="outline-text-6" id="text-orgb4e2845">
+<li><a id="org5cb6921"></a>Example :<br />
+<div class="outline-text-6" id="text-org5cb6921">
 <p>
 P(x) : sqrt(x) &gt; 0 ;  Q(x) : x ≥ 1
 </p>
@@ -860,8 +860,8 @@ P(x) : sqrt(x) &gt; 0 ;  Q(x) : x ≥ 1
 </li>
 </ul>
 </li>
-<li><a id="orgce6dd51"></a>Propriety Number 3:<br />
-<div class="outline-text-5" id="text-orgce6dd51">
+<li><a id="orgd56cb14"></a>Propriety Number 3:<br />
+<div class="outline-text-5" id="text-orgd56cb14">
 <p>
 <b>∃ x ∈ E, [P(x) ∧ Q(x)] <i>⇒</i> [∃ x ∈ E, P(x)] ∧ [∃ x ∈ E, Q(x)]</b>
 </p>
@@ -872,8 +872,8 @@ P(x) : sqrt(x) &gt; 0 ;  Q(x) : x ≥ 1
 </p>
 </div>
 <ul class="org-ul">
-<li><a id="org956d29b"></a>Example of why it&rsquo;s NOT an equivalence :<br />
-<div class="outline-text-6" id="text-org956d29b">
+<li><a id="org52c3098"></a>Example of why it&rsquo;s NOT an equivalence :<br />
+<div class="outline-text-6" id="text-org52c3098">
 <p>
 P(x) : x &gt; 5  ;  Q(x) : x &lt; 5
 </p>
@@ -886,8 +886,8 @@ 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="orgce400d9"></a>Propriety Number 4:<br />
-<div class="outline-text-5" id="text-orgce400d9">
+<li><a id="org9439534"></a>Propriety Number 4:<br />
+<div class="outline-text-5" id="text-org9439534">
 <p>
 <b>[∀ x ∈ E, P(x)] ∨ [∀ x ∈ E, Q(x)] <i>⇒</i> ∀x ∈ E, [P(x) ∨ Q(x)]</b>
 </p>
@@ -901,16 +901,16 @@ 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-org48fa1b7" class="outline-3">
-<h3 id="org48fa1b7">Multi-parameter proprieties :</h3>
-<div class="outline-text-3" id="text-org48fa1b7">
+<div id="outline-container-orgcb2ff75" class="outline-3">
+<h3 id="orgcb2ff75">Multi-parameter proprieties :</h3>
+<div class="outline-text-3" id="text-orgcb2ff75">
 <p>
 A propriety P can depend on two or more parameters, for convenience we call them x,y,z&#x2026;etc
 </p>
 </div>
 <ul class="org-ul">
-<li><a id="org985d3f3"></a>Example :<br />
-<div class="outline-text-6" id="text-org985d3f3">
+<li><a id="org309a152"></a>Example :<br />
+<div class="outline-text-6" id="text-org309a152">
 <p>
 P(x,y): x+y &gt; 0
 </p>
@@ -926,8 +926,8 @@ P(-2,-1) is a False one
 </p>
 </div>
 </li>
-<li><a id="orgd3167fe"></a>WARNING :<br />
-<div class="outline-text-6" id="text-orgd3167fe">
+<li><a id="orgfbf5cee"></a>WARNING :<br />
+<div class="outline-text-6" id="text-orgfbf5cee">
 <p>
 ∀x ∈ E, ∃y ∈ F , P(x,y)
 </p>
@@ -943,8 +943,8 @@ 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="orge81043c"></a>Example :<br />
-<div class="outline-text-7" id="text-orge81043c">
+<li><a id="org21332e2"></a>Example :<br />
+<div class="outline-text-7" id="text-org21332e2">
 <p>
 ∀ x ∈ ℕ , ∃ y ∈ ℕ y &gt; x -&#x2013;&#x2014; True
 </p>
@@ -958,8 +958,8 @@ Are different because in the first one y depends on x, while in the second one,
 </ul>
 </li>
 </ul>
-<li><a id="org96c2514"></a>Proprieties :<br />
-<div class="outline-text-5" id="text-org96c2514">
+<li><a id="org2fad1a6"></a>Proprieties :<br />
+<div class="outline-text-5" id="text-org2fad1a6">
 <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>
@@ -968,20 +968,20 @@ Are different because in the first one y depends on x, while in the second one,
 </li>
 </ul>
 </div>
-<div id="outline-container-org47ee190" class="outline-3">
-<h3 id="org47ee190">Methods of mathematical reasoning :</h3>
-<div class="outline-text-3" id="text-org47ee190">
+<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>
-<div id="outline-container-org24c7fa4" class="outline-4">
-<h4 id="org24c7fa4">Direct reasoning :</h4>
-<div class="outline-text-4" id="text-org24c7fa4">
+<div id="outline-container-org0e2120a" class="outline-4">
+<h4 id="org0e2120a">Direct reasoning :</h4>
+<div class="outline-text-4" id="text-org0e2120a">
 <p>
 To show that an implication P ⇒ Q is true, we suppose that P is true and we show that Q is true
 </p>
 </div>
 <ul class="org-ul">
-<li><a id="orgfa904f5"></a>Example:<br />
-<div class="outline-text-5" id="text-orgfa904f5">
+<li><a id="orge655791"></a>Example:<br />
+<div class="outline-text-5" id="text-orge655791">
 <p>
 Let a,b be two Real numbers, we have to prove that <b>a² + b² = 1 ⇒ |a + b| ≤ 2</b>
 </p>
@@ -1024,9 +1024,9 @@ a²+b²=1 ⇒ |a + b| ≤ 2 <b>Which is what we wanted to prove, therefor the im
 </li>
 </ul>
 </div>
-<div id="outline-container-orgb9d2c9e" class="outline-4">
-<h4 id="orgb9d2c9e">Reasoning by the Absurd:</h4>
-<div class="outline-text-4" id="text-orgb9d2c9e">
+<div id="outline-container-org3318c18" class="outline-4">
+<h4 id="org3318c18">Reasoning by the Absurd:</h4>
+<div class="outline-text-4" id="text-org3318c18">
 <p>
 To prove that a proposition is True, we suppose that it&rsquo;s False and we must come to a contradiction
 </p>
@@ -1037,8 +1037,8 @@ 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="orgfbfd0bb"></a>Example:<br />
-<div class="outline-text-5" id="text-orgfbfd0bb">
+<li><a id="org6217ba8"></a>Example:<br />
+<div class="outline-text-5" id="text-org6217ba8">
 <p>
 Prove that this proposition is correct using the reasoning by the absurd : ∀x ∈ ℝ* , sqrt(1+x²) ≠ 1 + x²/2
 </p>
@@ -1056,17 +1056,17 @@ 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-org8b209a6" class="outline-4">
-<h4 id="org8b209a6">Reasoning by contraposition:</h4>
-<div class="outline-text-4" id="text-org8b209a6">
+<div id="outline-container-orgdca4b33" class="outline-4">
+<h4 id="orgdca4b33">Reasoning by contraposition:</h4>
+<div class="outline-text-4" id="text-orgdca4b33">
 <p>
 If an implication P ⇒ Q is too hard to prove, we just have to prove not(Q) ⇒ not(P) is true !!! or in other words that both not(P) and not(Q) are true
 </p>
 </div>
 </div>
-<div id="outline-container-org94c9a28" class="outline-4">
-<h4 id="org94c9a28">Reasoning by counter example:</h4>
-<div class="outline-text-4" id="text-org94c9a28">
+<div id="outline-container-org45373bc" class="outline-4">
+<h4 id="org45373bc">Reasoning by counter example:</h4>
+<div class="outline-text-4" id="text-org45373bc">
 <p>
 To prove that a proposition ∀x ∈ E, P(x) is false, all we have to do is find a single value of x from E such as not(P(x)) is true
 </p>
@@ -1074,20 +1074,20 @@ 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-org0eabee1" class="outline-2">
-<h2 id="org0eabee1">3eme Cours : <i>Oct 9</i></h2>
-<div class="outline-text-2" id="text-org0eabee1">
+<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>
-<div id="outline-container-orgdafe6b7" class="outline-4">
-<h4 id="orgdafe6b7">Reasoning by recurrence :</h4>
-<div class="outline-text-4" id="text-orgdafe6b7">
+<div id="outline-container-orgc3cdd55" class="outline-4">
+<h4 id="orgc3cdd55">Reasoning by recurrence :</h4>
+<div class="outline-text-4" id="text-orgc3cdd55">
 <p>
 P is a propriety dependent of <b>n ∈ ℕ</b>. If for n0 ∈ ℕ P(n0) is true, and if for n ≥ n0 (P(n) ⇒ P(n+1)) is true. Then P(n) is true for n ≥ n0
 </p>
 </div>
 <ul class="org-ul">
-<li><a id="org9970baf"></a>Example:<br />
-<div class="outline-text-5" id="text-org9970baf">
+<li><a id="org74698a3"></a>Example:<br />
+<div class="outline-text-5" id="text-org74698a3">
 <p>
 Let&rsquo;s prove that ∀ n ≥ 1 , (n,k=1)Σk = [n(n+1)]/2
 </p>
@@ -1123,21 +1123,21 @@ 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-org4d1906f" class="outline-2">
-<h2 id="org4d1906f">4eme Cours : Chapitre 2 : Sets and Operations</h2>
-<div class="outline-text-2" id="text-org4d1906f">
+<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>
-<div id="outline-container-orgd19c38e" class="outline-3">
-<h3 id="orgd19c38e">Definition of a set :</h3>
-<div class="outline-text-3" id="text-orgd19c38e">
+<div id="outline-container-org5c29bea" class="outline-3">
+<h3 id="org5c29bea">Definition of a set :</h3>
+<div class="outline-text-3" id="text-org5c29bea">
 <p>
 A set is a collection of objects that share the sane propriety
 </p>
 </div>
 </div>
-<div id="outline-container-orgcf58c48" class="outline-3">
-<h3 id="orgcf58c48">Belonging, inclusion, and equality :</h3>
-<div class="outline-text-3" id="text-orgcf58c48">
+<div id="outline-container-org7f4934f" class="outline-3">
+<h3 id="org7f4934f">Belonging, inclusion, and equality :</h3>
+<div class="outline-text-3" id="text-org7f4934f">
 <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&rsquo;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>
@@ -1146,13 +1146,13 @@ A set is a collection of objects that share the sane propriety
 </ol>
 </div>
 </div>
-<div id="outline-container-org939fd93" class="outline-3">
-<h3 id="org939fd93">Intersections and reunions :</h3>
-<div class="outline-text-3" id="text-org939fd93">
+<div id="outline-container-orgd439312" class="outline-3">
+<h3 id="orgd439312">Intersections and reunions :</h3>
+<div class="outline-text-3" id="text-orgd439312">
 </div>
-<div id="outline-container-orge8ae0b6" class="outline-4">
-<h4 id="orge8ae0b6">Intersection:</h4>
-<div class="outline-text-4" id="text-orge8ae0b6">
+<div id="outline-container-org2eaf0a6" class="outline-4">
+<h4 id="org2eaf0a6">Intersection:</h4>
+<div class="outline-text-4" id="text-org2eaf0a6">
 <p>
 E ∩ F = {x / x ∈ E AND x ∈ F} ; x ∈ E ∩ F ⇔ x ∈ F AND x ∈ F
 </p>
@@ -1163,9 +1163,9 @@ x ∉ E ∩ F ⇔ x ∉ E OR x ∉ F
 </p>
 </div>
 </div>
-<div id="outline-container-org07c050a" class="outline-4">
-<h4 id="org07c050a">Union:</h4>
-<div class="outline-text-4" id="text-org07c050a">
+<div id="outline-container-org8bfbedf" class="outline-4">
+<h4 id="org8bfbedf">Union:</h4>
+<div class="outline-text-4" id="text-org8bfbedf">
 <p>
 E ∪ F = {x / x ∈ E OR x ∈ F} ;  x ∈ E ∪ F ⇔ x ∈ F OR x ∈ F
 </p>
@@ -1176,17 +1176,17 @@ x ∉ E ∪ F ⇔ x ∉ E AND x ∉ F
 </p>
 </div>
 </div>
-<div id="outline-container-org7ecf856" class="outline-4">
-<h4 id="org7ecf856">Difference between two sets:</h4>
-<div class="outline-text-4" id="text-org7ecf856">
+<div id="outline-container-orgf5d7c25" class="outline-4">
+<h4 id="orgf5d7c25">Difference between two sets:</h4>
+<div class="outline-text-4" id="text-orgf5d7c25">
 <p>
 E\F(Which is also written as : E - F) = {x / x ∈ E and x ∉ F}
 </p>
 </div>
 </div>
-<div id="outline-container-orgad5f4da" class="outline-4">
-<h4 id="orgad5f4da">Complimentary set:</h4>
-<div class="outline-text-4" id="text-orgad5f4da">
+<div id="outline-container-org16f26ee" class="outline-4">
+<h4 id="org16f26ee">Complimentary set:</h4>
+<div class="outline-text-4" id="text-org16f26ee">
 <p>
 If F ⊂ E. E - F is the complimentary of F in E.
 </p>
@@ -1197,52 +1197,52 @@ FCE = {x /x ∈ E AND x ∉ F} <b>ONLY WHEN F IS A SUBSET OF E</b>
 </p>
 </div>
 </div>
-<div id="outline-container-org3e8e3b3" class="outline-4">
-<h4 id="org3e8e3b3">Symentrical difference</h4>
-<div class="outline-text-4" id="text-org3e8e3b3">
+<div id="outline-container-org67da9c0" class="outline-4">
+<h4 id="org67da9c0">Symentrical difference</h4>
+<div class="outline-text-4" id="text-org67da9c0">
 <p>
 E Δ F = (E - F) ∪ (F - E) ; = (E ∪ F) - (E ∩ F)
 </p>
 </div>
 </div>
 </div>
-<div id="outline-container-org8920c77" class="outline-3">
-<h3 id="org8920c77">Proprieties :</h3>
-<div class="outline-text-3" id="text-org8920c77">
+<div id="outline-container-org10858f6" class="outline-3">
+<h3 id="org10858f6">Proprieties :</h3>
+<div class="outline-text-3" id="text-org10858f6">
 <p>
 Let E,F and G be 3 sets. We have :
 </p>
 </div>
-<div id="outline-container-orgcb406ce" class="outline-4">
-<h4 id="orgcb406ce">Commutativity:</h4>
-<div class="outline-text-4" id="text-orgcb406ce">
+<div id="outline-container-orgdeeff37" class="outline-4">
+<h4 id="orgdeeff37">Commutativity:</h4>
+<div class="outline-text-4" id="text-orgdeeff37">
 <p>
 E ∩ F = F ∩ E
 E ∪ F = F ∪ E
 </p>
 </div>
 </div>
-<div id="outline-container-orgfcaf63a" class="outline-4">
-<h4 id="orgfcaf63a">Associativity:</h4>
-<div class="outline-text-4" id="text-orgfcaf63a">
+<div id="outline-container-org6228f00" class="outline-4">
+<h4 id="org6228f00">Associativity:</h4>
+<div class="outline-text-4" id="text-org6228f00">
 <p>
 E ∩ (F ∩ G) = (E ∩ F) ∩ G
 E ∪ (F ∪ G) = (E ∪ F) ∪ G
 </p>
 </div>
 </div>
-<div id="outline-container-org6ad9182" class="outline-4">
-<h4 id="org6ad9182">Distributivity:</h4>
-<div class="outline-text-4" id="text-org6ad9182">
+<div id="outline-container-org2523e0e" class="outline-4">
+<h4 id="org2523e0e">Distributivity:</h4>
+<div class="outline-text-4" id="text-org2523e0e">
 <p>
 E ∩ (F ∪ G) = (E ∩ F) ∪ (E ∩ G)
 E ∪ (F ∩ G) = (E ∪ F) ∩ (E ∪ G)
 </p>
 </div>
 </div>
-<div id="outline-container-org7a0450e" class="outline-4">
-<h4 id="org7a0450e">Lois de Morgan:</h4>
-<div class="outline-text-4" id="text-org7a0450e">
+<div id="outline-container-orgeb0c0a3" class="outline-4">
+<h4 id="orgeb0c0a3">Lois de Morgan:</h4>
+<div class="outline-text-4" id="text-orgeb0c0a3">
 <p>
 If E ⊂ G and F ⊂ G ;
 </p>
@@ -1252,33 +1252,33 @@ If E ⊂ G and F ⊂ G ;
 </p>
 </div>
 </div>
-<div id="outline-container-org44fd147" class="outline-4">
-<h4 id="org44fd147">An other one:</h4>
-<div class="outline-text-4" id="text-org44fd147">
+<div id="outline-container-orge638501" class="outline-4">
+<h4 id="orge638501">An other one:</h4>
+<div class="outline-text-4" id="text-orge638501">
 <p>
 E - (F ∩ G) = (E-F) ∪ (E-G) ;  E - (F ∪ G) = (E-F) ∩ (E-G)
 </p>
 </div>
 </div>
-<div id="outline-container-orgca3a4c6" class="outline-4">
-<h4 id="orgca3a4c6">An other one:</h4>
-<div class="outline-text-4" id="text-orgca3a4c6">
+<div id="outline-container-orgfe0b562" class="outline-4">
+<h4 id="orgfe0b562">An other one:</h4>
+<div class="outline-text-4" id="text-orgfe0b562">
 <p>
 E ∩ ∅ = ∅ ; E ∪ ∅ = E
 </p>
 </div>
 </div>
-<div id="outline-container-org6cd18a3" class="outline-4">
-<h4 id="org6cd18a3">And an other one:</h4>
-<div class="outline-text-4" id="text-org6cd18a3">
+<div id="outline-container-org48afea2" class="outline-4">
+<h4 id="org48afea2">And an other one:</h4>
+<div class="outline-text-4" id="text-org48afea2">
 <p>
 E ∩ (F Δ G) = (E ∩ F) Δ (E ∩ G)
 </p>
 </div>
 </div>
-<div id="outline-container-org0889163" class="outline-4">
-<h4 id="org0889163">And the last one:</h4>
-<div class="outline-text-4" id="text-org0889163">
+<div id="outline-container-org1138be8" class="outline-4">
+<h4 id="org1138be8">And the last one:</h4>
+<div class="outline-text-4" id="text-org1138be8">
 <p>
 E Δ ∅ = E ; E Δ E = ∅
 </p>
@@ -1286,10 +1286,167 @@ E Δ ∅ = E ; E Δ E = ∅
 </div>
 </div>
 </div>
+<div id="outline-container-orgf188863" class="outline-2">
+<h2 id="orgf188863">5eme cours: L&rsquo;ensemble des parties d&rsquo;un ensemble <i>Oct 16</i></h2>
+<div class="outline-text-2" id="text-orgf188863">
+<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>
+</p>
+</div>
+<div id="outline-container-org6cfe0d7" class="outline-4">
+<h4 id="org6cfe0d7">Notes :</h4>
+<div class="outline-text-4" id="text-org6cfe0d7">
+<p>
+∅ ∈ P(E) ; E ∈ P(E)
+</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>
+</p>
+</div>
+</div>
+<div id="outline-container-orgd0b341d" class="outline-4">
+<h4 id="orgd0b341d">Examples :</h4>
+<div class="outline-text-4" id="text-orgd0b341d">
+<p>
+E = {a,b,c} // P(E)={∅, {a}, {b}, {c}, {a,b}, {b,c}, {a,c}, {a,b,c}}
+</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">
+<p>
+We say that <b>A</b> is a partition of E if:
+</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>
+</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">
+<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
+</p>
+</div>
+<div id="outline-container-org4b0f328" class="outline-4">
+<h4 id="org4b0f328">Example :</h4>
+<div class="outline-text-4" id="text-org4b0f328">
+<p>
+A = {4,5} ; B= {4,5,6} // AxB = {(4,4), (4,5), (4,6), (5,4), (5,5), (5,6)}
+</p>
+
+
+<p>
+BxA = {(4,4), (4,5), (5,4), (5,5), (6,4), (6,5)} // Therefore AxB ≠ BxA
+</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">
+<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>
+</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>
+<div id="outline-container-orgeb6cba6" class="outline-3">
+<h3 id="orgeb6cba6">Definition :</h3>
+<div class="outline-text-3" id="text-orgeb6cba6">
+<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>
+</p>
+</div>
+</div>
+<div id="outline-container-org1696189" class="outline-3">
+<h3 id="org1696189">Proprieties :</h3>
+<div class="outline-text-3" id="text-org1696189">
+<p>
+Let E be a set and R a relation defined in E
+</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>
+</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">
+<p>
+We say that R is a relation of equivalence in E if its reflexive, symetrical and transitive
+</p>
+</div>
+<div id="outline-container-org110e6fa" class="outline-4">
+<h4 id="org110e6fa">Equivalence class :</h4>
+<div class="outline-text-4" id="text-org110e6fa">
+<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 :
+</p>
+
+
+<p>
+<b>a̅ = {y ∈ E/ y R a}</b>
+</p>
+</div>
+<ul class="org-ul">
+<li><a id="org20e3b3b"></a>The quotient set :<br />
+<div class="outline-text-5" id="text-org20e3b3b">
+<p>
+E/R = {̅a , a ∈ E}
+</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">
+<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.
+</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>
+</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">
+<p>
+∀x,y ∈ ℝ , xRy ⇔ x²-y²=x-y
+</p>
+<ol class="org-ol">
+<li>Prove that R is an equivalence relation</li>
+<li>Let a ∈ ℝ, find ̅a</li>
+</ol>
+</div>
+</div>
+</div>
+</div>
 </div>
 <div id="postamble" class="status">
 <p class="author">Author: Crystal</p>
-<p class="date">Created: 2023-10-13 Fri 16:58</p>
+<p class="date">Created: 2023-10-17 Tue 22:32</p>
 </div>
 </body>
 </html>
\ No newline at end of file
author	Crystal <crystal@wizard.tower>	2023-10-17 22:42:30 +0100
committer	Crystal <crystal@wizard.tower>	2023-10-17 22:42:30 +0100
commit	16e5f6e90b603bc62d4c0af24d79ca5e6121ac6a (patch)
tree	b8ea1a57004ec9498fd41988b84f355deb2fff78
parent	7315416eaee4c833bff711de48975ed95bedf3fd (diff)
download	www-16e5f6e90b603bc62d4c0af24d79ca5e6121ac6a.tar.gz