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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’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’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 & 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 & a disjunction</b>:</h4> +<div class="outline-text-4" id="text-org7422610"> <p> Now we won’t use bars here because my lazy ass doesn’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 < 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 > 5 ⇔ ∃ x ≥ 1 x² < 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 “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="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) > 0 ; Q(x) : x ≥ 1 </p> @@ -860,8 +860,8 @@ P(x) : sqrt(x) > 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) > 0 ; Q(x) : x ≥ 1 </p> </div> <ul class="org-ul"> -<li><a id="org956d29b"></a>Example of why it’s NOT an equivalence :<br /> -<div class="outline-text-6" id="text-org956d29b"> +<li><a id="org52c3098"></a>Example of why it’s NOT an equivalence :<br /> +<div class="outline-text-6" id="text-org52c3098"> <p> P(x) : x > 5 ; Q(x) : x < 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…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 > 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 > x -–— 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’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’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’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’ensemble des parties d’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 |