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diff --git a/uni_notes/algebra.html b/uni_notes/algebra.html index 27c8d76..9323119 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> -<!-- 2023-10-11 Wed 19:04 --> +<!-- 2023-10-13 Fri 16:58 --> <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-orgcc5f4d8" class="outline-2"> -<h2 id="orgcc5f4d8">Contenu de la Matiére</h2> -<div class="outline-text-2" id="text-orgcc5f4d8"> +<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> -<div id="outline-container-orgf4040f2" class="outline-3"> -<h3 id="orgf4040f2">Rappels et compléments (11H)</h3> -<div class="outline-text-3" id="text-orgf4040f2"> +<div id="outline-container-orgae00938" class="outline-3"> +<h3 id="orgae00938">Rappels et compléments (11H)</h3> +<div class="outline-text-3" id="text-orgae00938"> <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-orge74cdc9" class="outline-3"> -<h3 id="orge74cdc9">Structures Algébriques (11H)</h3> -<div class="outline-text-3" id="text-orge74cdc9"> +<div id="outline-container-org0eb35c9" class="outline-3"> +<h3 id="org0eb35c9">Structures Algébriques (11H)</h3> +<div class="outline-text-3" id="text-org0eb35c9"> <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-org0b54650" class="outline-3"> -<h3 id="org0b54650">Polynômes et fractions rationnelles</h3> -<div class="outline-text-3" id="text-org0b54650"> +<div id="outline-container-org4a088f6" class="outline-3"> +<h3 id="org4a088f6">Polynômes et fractions rationnelles</h3> +<div class="outline-text-3" id="text-org4a088f6"> <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-org9dbc8bb" class="outline-2"> -<h2 id="org9dbc8bb">Premier cours : Logique mathématique et méthodes du raisonnement mathématique <i>Sep 25</i> :</h2> -<div class="outline-text-2" id="text-org9dbc8bb"> +<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"> <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-org29099d5" class="outline-3"> -<h3 id="org29099d5">Properties:</h3> -<div class="outline-text-3" id="text-org29099d5"> +<div id="outline-container-org74d9557" class="outline-3"> +<h3 id="org74d9557">Properties:</h3> +<div class="outline-text-3" id="text-org74d9557"> </div> -<div id="outline-container-orgfeb62c8" class="outline-4"> -<h4 id="orgfeb62c8"><b>Absorption</b>:</h4> -<div class="outline-text-4" id="text-orgfeb62c8"> +<div id="outline-container-orgc7f1d03" class="outline-4"> +<h4 id="orgc7f1d03"><b>Absorption</b>:</h4> +<div class="outline-text-4" id="text-orgc7f1d03"> <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-org2fa2b1f" class="outline-4"> -<h4 id="org2fa2b1f"><b>Commutativity</b>:</h4> -<div class="outline-text-4" id="text-org2fa2b1f"> +<div id="outline-container-orgcb729de" class="outline-4"> +<h4 id="orgcb729de"><b>Commutativity</b>:</h4> +<div class="outline-text-4" id="text-orgcb729de"> <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-orge30be26" class="outline-4"> -<h4 id="orge30be26"><b>Associativity</b>:</h4> -<div class="outline-text-4" id="text-orge30be26"> +<div id="outline-container-org4ae8933" class="outline-4"> +<h4 id="org4ae8933"><b>Associativity</b>:</h4> +<div class="outline-text-4" id="text-org4ae8933"> <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-orgd1de8d5" class="outline-4"> -<h4 id="orgd1de8d5"><b>Distributivity</b>:</h4> -<div class="outline-text-4" id="text-orgd1de8d5"> +<div id="outline-container-org095f4a6" class="outline-4"> +<h4 id="org095f4a6"><b>Distributivity</b>:</h4> +<div class="outline-text-4" id="text-org095f4a6"> <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-org8c5c9f6" class="outline-4"> -<h4 id="org8c5c9f6"><b>Neutral element</b>:</h4> -<div class="outline-text-4" id="text-org8c5c9f6"> +<div id="outline-container-orgf7e29ee" class="outline-4"> +<h4 id="orgf7e29ee"><b>Neutral element</b>:</h4> +<div class="outline-text-4" id="text-orgf7e29ee"> <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-orgf102dde" class="outline-4"> -<h4 id="orgf102dde"><b>Negation of a conjunction & a disjunction</b>:</h4> -<div class="outline-text-4" id="text-orgf102dde"> +<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"> <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-org9534ece" class="outline-4"> -<h4 id="org9534ece"><b>Transitivity</b>:</h4> -<div class="outline-text-4" id="text-org9534ece"> +<div id="outline-container-orgd144fb3" class="outline-4"> +<h4 id="orgd144fb3"><b>Transitivity</b>:</h4> +<div class="outline-text-4" id="text-orgd144fb3"> <p> -[(P ⇒ Q) (Q ⇒ R)] ⇔ P ⇒ R +[(P ⇒ Q) AND (Q ⇒ R)] ⇔ P ⇒ R </p> </div> </div> -<div id="outline-container-org90fa987" class="outline-4"> -<h4 id="org90fa987"><b>Contraposition</b>:</h4> -<div class="outline-text-4" id="text-org90fa987"> +<div id="outline-container-orgbc26f01" class="outline-4"> +<h4 id="orgbc26f01"><b>Contraposition</b>:</h4> +<div class="outline-text-4" id="text-orgbc26f01"> <p> (P ⇒ Q) ⇔ (Q̅ ⇒ P̅) </p> </div> </div> -<div id="outline-container-orga2d0ece" class="outline-4"> -<h4 id="orga2d0ece">God only knows what this property is called:</h4> -<div class="outline-text-4" id="text-orga2d0ece"> +<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"> <p> <i>If</i> </p> @@ -558,7 +558,7 @@ and </p> <p> -(Q̅ ⇒ Q) is true +(P̅ ⇒ Q) is true </p> <p> @@ -571,17 +571,17 @@ Q is always true </div> </div> </div> -<div id="outline-container-org35a43b7" class="outline-3"> -<h3 id="org35a43b7">Some exercices I found online :</h3> -<div class="outline-text-3" id="text-org35a43b7"> +<div 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> -<div id="outline-container-orgf619324" class="outline-4"> -<h4 id="orgf619324">USTHB 2022/2023 Section B :</h4> -<div class="outline-text-4" id="text-orgf619324"> +<div 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> <ul class="org-ul"> -<li><a id="org1c47389"></a>Exercice 1: Démontrer les équivalences suivantes:<br /> -<div class="outline-text-5" id="text-org1c47389"> +<li><a id="orgdcdfa08"></a>Exercice 1: Démontrer les équivalences suivantes:<br /> +<div class="outline-text-5" id="text-orgdcdfa08"> <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="org688fdcc"></a>Exercice 2: Dire si les propositions suivantes sont vraies ou fausses, et les nier:<br /> -<div class="outline-text-5" id="text-org688fdcc"> +<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"> <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-orgac834f2" class="outline-2"> -<h2 id="orgac834f2">2éme cours <i>Oct 2</i></h2> -<div class="outline-text-2" id="text-orgac834f2"> +<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> -<div id="outline-container-org42a8fad" class="outline-3"> -<h3 id="org42a8fad">Quantifiers</h3> -<div class="outline-text-3" id="text-org42a8fad"> +<div id="outline-container-org1d4ffa3" class="outline-3"> +<h3 id="org1d4ffa3">Quantifiers</h3> +<div class="outline-text-3" id="text-org1d4ffa3"> <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="org48c4773"></a>Example<br /> -<div class="outline-text-6" id="text-org48c4773"> +<li><a id="orgd0b7f53"></a>Example<br /> +<div class="outline-text-6" id="text-orgd0b7f53"> <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-org923b5fe" class="outline-4"> -<h4 id="org923b5fe">Proprieties</h4> -<div class="outline-text-4" id="text-org923b5fe"> +<div id="outline-container-orga7cd185" class="outline-4"> +<h4 id="orga7cd185">Proprieties</h4> +<div class="outline-text-4" id="text-orga7cd185"> </div> <ul class="org-ul"> -<li><a id="orgaf93dd2"></a>Propriety Number 1:<br /> -<div class="outline-text-5" id="text-orgaf93dd2"> +<li><a id="org7460082"></a>Propriety Number 1:<br /> +<div class="outline-text-5" id="text-org7460082"> <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="orgf4f038f"></a>Example:<br /> -<div class="outline-text-6" id="text-orgf4f038f"> +<li><a id="org27b8375"></a>Example:<br /> +<div class="outline-text-6" id="text-org27b8375"> <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="org2c00dae"></a>Propriety Number 2:<br /> -<div class="outline-text-5" id="text-org2c00dae"> +<li><a id="org21aa647"></a>Propriety Number 2:<br /> +<div class="outline-text-5" id="text-org21aa647"> <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="orga95ad05"></a>Example :<br /> -<div class="outline-text-6" id="text-orga95ad05"> +<li><a id="orgb4e2845"></a>Example :<br /> +<div class="outline-text-6" id="text-orgb4e2845"> <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="orgfe5ddf2"></a>Propriety Number 3:<br /> -<div class="outline-text-5" id="text-orgfe5ddf2"> +<li><a id="orgce6dd51"></a>Propriety Number 3:<br /> +<div class="outline-text-5" id="text-orgce6dd51"> <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="orgca7bea7"></a>Example of why it’s NOT an equivalence :<br /> -<div class="outline-text-6" id="text-orgca7bea7"> +<li><a id="org956d29b"></a>Example of why it’s NOT an equivalence :<br /> +<div class="outline-text-6" id="text-org956d29b"> <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="orgf4ecdc0"></a>Propriety Number 4:<br /> -<div class="outline-text-5" id="text-orgf4ecdc0"> +<li><a id="orgce400d9"></a>Propriety Number 4:<br /> +<div class="outline-text-5" id="text-orgce400d9"> <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-org6421557" class="outline-3"> -<h3 id="org6421557">Multi-parameter proprieties :</h3> -<div class="outline-text-3" id="text-org6421557"> +<div id="outline-container-org48fa1b7" class="outline-3"> +<h3 id="org48fa1b7">Multi-parameter proprieties :</h3> +<div class="outline-text-3" id="text-org48fa1b7"> <p> A propriety P can depend on two or more parameters, for convenience we call them x,y,z…etc </p> </div> <ul class="org-ul"> -<li><a id="org314cff3"></a>Example :<br /> -<div class="outline-text-6" id="text-org314cff3"> +<li><a id="org985d3f3"></a>Example :<br /> +<div class="outline-text-6" id="text-org985d3f3"> <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="orga9b6089"></a>WARNING :<br /> -<div class="outline-text-6" id="text-orga9b6089"> +<li><a id="orgd3167fe"></a>WARNING :<br /> +<div class="outline-text-6" id="text-orgd3167fe"> <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="org3c5e5a8"></a>Example :<br /> -<div class="outline-text-7" id="text-org3c5e5a8"> +<li><a id="orge81043c"></a>Example :<br /> +<div class="outline-text-7" id="text-orge81043c"> <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="org2f3208e"></a>Proprieties :<br /> -<div class="outline-text-5" id="text-org2f3208e"> +<li><a id="org96c2514"></a>Proprieties :<br /> +<div class="outline-text-5" id="text-org96c2514"> <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-orgc25dd7a" class="outline-3"> -<h3 id="orgc25dd7a">Methods of mathematical reasoning :</h3> -<div class="outline-text-3" id="text-orgc25dd7a"> +<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> -<div id="outline-container-orgf843851" class="outline-4"> -<h4 id="orgf843851">Direct reasoning :</h4> -<div class="outline-text-4" id="text-orgf843851"> +<div id="outline-container-org24c7fa4" class="outline-4"> +<h4 id="org24c7fa4">Direct reasoning :</h4> +<div class="outline-text-4" id="text-org24c7fa4"> <p> To show that an implication P ⇒ Q is true, we suppose that P is true and we show that Q is true </p> </div> <ul class="org-ul"> -<li><a id="org6dd2136"></a>Example:<br /> -<div class="outline-text-5" id="text-org6dd2136"> +<li><a id="orgfa904f5"></a>Example:<br /> +<div class="outline-text-5" id="text-orgfa904f5"> <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-orga59c0ad" class="outline-4"> -<h4 id="orga59c0ad">Reasoning by the Absurd:</h4> -<div class="outline-text-4" id="text-orga59c0ad"> +<div id="outline-container-orgb9d2c9e" class="outline-4"> +<h4 id="orgb9d2c9e">Reasoning by the Absurd:</h4> +<div class="outline-text-4" id="text-orgb9d2c9e"> <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="orgf8b9f83"></a>Example:<br /> -<div class="outline-text-5" id="text-orgf8b9f83"> +<li><a id="orgfbfd0bb"></a>Example:<br /> +<div class="outline-text-5" id="text-orgfbfd0bb"> <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-orgcc285c2" class="outline-4"> -<h4 id="orgcc285c2">Reasoning by contraposition:</h4> -<div class="outline-text-4" id="text-orgcc285c2"> +<div id="outline-container-org8b209a6" class="outline-4"> +<h4 id="org8b209a6">Reasoning by contraposition:</h4> +<div class="outline-text-4" id="text-org8b209a6"> <p> If an implication P ⇒ Q is too hard to prove, we just have to prove not(Q) ⇒ not(P) is true !!! or in other words that both not(P) and not(Q) are true </p> </div> </div> -<div id="outline-container-org2e67808" class="outline-4"> -<h4 id="org2e67808">Reasoning by counter example:</h4> -<div class="outline-text-4" id="text-org2e67808"> +<div id="outline-container-org94c9a28" class="outline-4"> +<h4 id="org94c9a28">Reasoning by counter example:</h4> +<div class="outline-text-4" id="text-org94c9a28"> <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-org7440601" class="outline-2"> -<h2 id="org7440601">3eme Cours : <i>Oct 9</i></h2> -<div class="outline-text-2" id="text-org7440601"> +<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> -<div id="outline-container-org70aa2db" class="outline-4"> -<h4 id="org70aa2db">Reasoning by recurrence :</h4> -<div class="outline-text-4" id="text-org70aa2db"> +<div id="outline-container-orgdafe6b7" class="outline-4"> +<h4 id="orgdafe6b7">Reasoning by recurrence :</h4> +<div class="outline-text-4" id="text-orgdafe6b7"> <p> P is a propriety dependent of <b>n ∈ ℕ</b>. If for n0 ∈ ℕ P(n0) is true, and if for n ≥ n0 (P(n) ⇒ P(n+1)) is true. Then P(n) is true for n ≥ n0 </p> </div> <ul class="org-ul"> -<li><a id="org52e5298"></a>Example:<br /> -<div class="outline-text-5" id="text-org52e5298"> +<li><a id="org9970baf"></a>Example:<br /> +<div class="outline-text-5" id="text-org9970baf"> <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-orga6a518d" class="outline-2"> -<h2 id="orga6a518d">4eme Cours : Chapitre 2 : Sets and Operations</h2> -<div class="outline-text-2" id="text-orga6a518d"> +<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> -<div id="outline-container-org31e3615" class="outline-3"> -<h3 id="org31e3615">Definition of a set :</h3> -<div class="outline-text-3" id="text-org31e3615"> +<div id="outline-container-orgd19c38e" class="outline-3"> +<h3 id="orgd19c38e">Definition of a set :</h3> +<div class="outline-text-3" id="text-orgd19c38e"> <p> A set is a collection of objects that share the sane propriety </p> </div> </div> -<div id="outline-container-orgfa9bfd1" class="outline-3"> -<h3 id="orgfa9bfd1">Belonging, inclusion, and equality :</h3> -<div class="outline-text-3" id="text-orgfa9bfd1"> +<div id="outline-container-orgcf58c48" class="outline-3"> +<h3 id="orgcf58c48">Belonging, inclusion, and equality :</h3> +<div class="outline-text-3" id="text-orgcf58c48"> <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-org2a19707" class="outline-3"> -<h3 id="org2a19707">Intersections and reunions :</h3> -<div class="outline-text-3" id="text-org2a19707"> +<div id="outline-container-org939fd93" class="outline-3"> +<h3 id="org939fd93">Intersections and reunions :</h3> +<div class="outline-text-3" id="text-org939fd93"> </div> -<div id="outline-container-org6a5f566" class="outline-4"> -<h4 id="org6a5f566">Intersection:</h4> -<div class="outline-text-4" id="text-org6a5f566"> +<div id="outline-container-orge8ae0b6" class="outline-4"> +<h4 id="orge8ae0b6">Intersection:</h4> +<div class="outline-text-4" id="text-orge8ae0b6"> <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-org9bc9aeb" class="outline-4"> -<h4 id="org9bc9aeb">Union:</h4> -<div class="outline-text-4" id="text-org9bc9aeb"> +<div id="outline-container-org07c050a" class="outline-4"> +<h4 id="org07c050a">Union:</h4> +<div class="outline-text-4" id="text-org07c050a"> <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-org9a7f719" class="outline-4"> -<h4 id="org9a7f719">Difference between two sets:</h4> -<div class="outline-text-4" id="text-org9a7f719"> +<div id="outline-container-org7ecf856" class="outline-4"> +<h4 id="org7ecf856">Difference between two sets:</h4> +<div class="outline-text-4" id="text-org7ecf856"> <p> E\F(Which is also written as : E - F) = {x / x ∈ E and x ∉ F} </p> </div> </div> -<div id="outline-container-org5f5c721" class="outline-4"> -<h4 id="org5f5c721">Complimentary set:</h4> -<div class="outline-text-4" id="text-org5f5c721"> +<div id="outline-container-orgad5f4da" class="outline-4"> +<h4 id="orgad5f4da">Complimentary set:</h4> +<div class="outline-text-4" id="text-orgad5f4da"> <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-orga285d1d" class="outline-4"> -<h4 id="orga285d1d">Symentrical difference</h4> -<div class="outline-text-4" id="text-orga285d1d"> +<div id="outline-container-org3e8e3b3" class="outline-4"> +<h4 id="org3e8e3b3">Symentrical difference</h4> +<div class="outline-text-4" id="text-org3e8e3b3"> <p> E Δ F = (E - F) ∪ (F - E) ; = (E ∪ F) - (E ∩ F) </p> </div> </div> </div> -<div id="outline-container-orgc12e73b" class="outline-3"> -<h3 id="orgc12e73b">Proprieties :</h3> -<div class="outline-text-3" id="text-orgc12e73b"> +<div id="outline-container-org8920c77" class="outline-3"> +<h3 id="org8920c77">Proprieties :</h3> +<div class="outline-text-3" id="text-org8920c77"> <p> Let E,F and G be 3 sets. We have : </p> </div> -<div id="outline-container-org31d8697" class="outline-4"> -<h4 id="org31d8697">Commutativity:</h4> -<div class="outline-text-4" id="text-org31d8697"> +<div id="outline-container-orgcb406ce" class="outline-4"> +<h4 id="orgcb406ce">Commutativity:</h4> +<div class="outline-text-4" id="text-orgcb406ce"> <p> E ∩ F = F ∩ E E ∪ F = F ∪ E </p> </div> </div> -<div id="outline-container-org7080d99" class="outline-4"> -<h4 id="org7080d99">Associativity:</h4> -<div class="outline-text-4" id="text-org7080d99"> +<div id="outline-container-orgfcaf63a" class="outline-4"> +<h4 id="orgfcaf63a">Associativity:</h4> +<div class="outline-text-4" id="text-orgfcaf63a"> <p> E ∩ (F ∩ G) = (E ∩ F) ∩ G E ∪ (F ∪ G) = (E ∪ F) ∪ G </p> </div> </div> -<div id="outline-container-org13da04d" class="outline-4"> -<h4 id="org13da04d">Distributivity:</h4> -<div class="outline-text-4" id="text-org13da04d"> +<div id="outline-container-org6ad9182" class="outline-4"> +<h4 id="org6ad9182">Distributivity:</h4> +<div class="outline-text-4" id="text-org6ad9182"> <p> E ∩ (F ∪ G) = (E ∩ F) ∪ (E ∩ G) E ∪ (F ∩ G) = (E ∪ F) ∩ (E ∪ G) </p> </div> </div> -<div id="outline-container-orgaa33b71" class="outline-4"> -<h4 id="orgaa33b71">Lois de Morgan:</h4> -<div class="outline-text-4" id="text-orgaa33b71"> +<div id="outline-container-org7a0450e" class="outline-4"> +<h4 id="org7a0450e">Lois de Morgan:</h4> +<div class="outline-text-4" id="text-org7a0450e"> <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-org7e7db42" class="outline-4"> -<h4 id="org7e7db42">An other one:</h4> -<div class="outline-text-4" id="text-org7e7db42"> +<div id="outline-container-org44fd147" class="outline-4"> +<h4 id="org44fd147">An other one:</h4> +<div class="outline-text-4" id="text-org44fd147"> <p> E - (F ∩ G) = (E-F) ∪ (E-G) ; E - (F ∪ G) = (E-F) ∩ (E-G) </p> </div> </div> -<div id="outline-container-orgd02bd7f" class="outline-4"> -<h4 id="orgd02bd7f">An other one:</h4> -<div class="outline-text-4" id="text-orgd02bd7f"> +<div id="outline-container-orgca3a4c6" class="outline-4"> +<h4 id="orgca3a4c6">An other one:</h4> +<div class="outline-text-4" id="text-orgca3a4c6"> <p> E ∩ ∅ = ∅ ; E ∪ ∅ = E </p> </div> </div> -<div id="outline-container-org99eb39a" class="outline-4"> -<h4 id="org99eb39a">And an other one:</h4> -<div class="outline-text-4" id="text-org99eb39a"> +<div id="outline-container-org6cd18a3" class="outline-4"> +<h4 id="org6cd18a3">And an other one:</h4> +<div class="outline-text-4" id="text-org6cd18a3"> <p> E ∩ (F Δ G) = (E ∩ F) Δ (E ∩ G) </p> </div> </div> -<div id="outline-container-org3e9b2ef" class="outline-4"> -<h4 id="org3e9b2ef">And the last one:</h4> -<div class="outline-text-4" id="text-org3e9b2ef"> +<div id="outline-container-org0889163" class="outline-4"> +<h4 id="org0889163">And the last one:</h4> +<div class="outline-text-4" id="text-org0889163"> <p> E Δ ∅ = E ; E Δ E = ∅ </p> @@ -1289,7 +1289,7 @@ E Δ ∅ = E ; E Δ E = ∅ </div> <div id="postamble" class="status"> <p class="author">Author: Crystal</p> -<p class="date">Created: 2023-10-11 Wed 19:04</p> +<p class="date">Created: 2023-10-13 Fri 16:58</p> </div> </body> </html> \ No newline at end of file |