Algebra 1

Contenu de la Matiére

Rappels et compléments (11H)

Logique mathématique et méthodes du raisonnement mathématique
Ensembles et Relations

Structures Algébriques (11H)

Groupes et morphisme de groupes
Anneaux et morphisme d’anneaux

Polynômes et fractions rationnelles

Notion du polynôme à une indéterminée á coefficients dans un anneau
Opérations Algébriques sur les polynômes

Premier cours : Logique mathématique et méthodes du raisonnement mathématique Sep 25 :

Let P Q and R be propositions which can either be True or False. And let’s also give the value 1 to each True proposition and 0 to each false one.

@@ -438,13 +438,13 @@ A proposition is equivalent to another only when both of them have the same v Note: P implying Q is equivalent to P̅ implying Q̅, or: (P ⇒ Q) ⇔ (P̅ ⇒ Q̅)

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-
Properties:
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Properties:
+

-
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Absorption:
-
+
+
Absorption:
+

(P ∨ P) ⇔ P
@@ -454,9 +454,9 @@ A proposition is equivalent to another only when both of them have the same v

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-
Commutativity:
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Commutativity:
+

(P ∧ Q) ⇔ (Q ∧ P)
@@ -466,9 +466,9 @@ A proposition is equivalent to another only when both of them have the same v

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Associativity:
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Associativity:
+

P ∧ (Q ∧ R) ⇔ (P ∧ Q) ∧ R
@@ -478,9 +478,9 @@ P ∨ (Q ∨ R) ⇔ (P ∨ Q) ∨ R

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-
Distributivity:
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+
Distributivity:
+

P ∧ (Q ∨ R) ⇔ (P ∧ Q) ∨ (P ∧ R)
@@ -490,9 +490,9 @@ P ∨ (Q ∧ R) ⇔ (P ∨ Q) ∧ (P ∨ R)

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Neutral element:
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Neutral element:
+

We define proposition T to be always true and F to be always false
@@ -506,9 +506,9 @@ P ∨ F ⇔ P

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Negation of a conjunction & a disjunction:
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Negation of a conjunction & a disjunction:
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Now we won’t use bars here because my lazy ass doesn’t know how, so instead I will use not()!!!
@@ -526,25 +526,25 @@ not(P ∨ Q) ⇔ P̅ ∧ Q̅

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Transitivity:
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Transitivity:
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[(P ⇒ Q) AND (Q ⇒ R)] ⇔ P ⇒ R

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Contraposition:
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Contraposition:
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(P ⇒ Q) ⇔ (Q̅ ⇒ P̅)

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God only knows what this property is called:
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+
God only knows what this property is called:
+

If
@@ -571,17 +571,17 @@ Q is always true

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Some exercices I found online :
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Some exercices I found online :
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USTHB 2022/2023 Section B :
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+
USTHB 2022/2023 Section B :
+

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Exercice 1: Démontrer les équivalences suivantes:
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Exercice 1: Démontrer les équivalences suivantes:
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(P ⇒ Q) ⇔ (Q̅ ⇒ P̅) @@ -635,8 +635,8 @@ Literally the same as above 🩷

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Exercice 2: Dire si les propositions suivantes sont vraies ou fausses, et les nier:
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Exercice 2: Dire si les propositions suivantes sont vraies ou fausses, et les nier:
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∀x ∈ ℝ ,∃y ∈ ℝ*+, tels que e^x = y @@ -769,13 +769,13 @@ y + x < 8

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2éme cours Oct 2
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+
2éme cours Oct 2
+

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Quantifiers
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Quantifiers
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A propriety P can depend on a parameter x
@@ -791,8 +791,8 @@ A propriety P can depend on a parameter x

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Example
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Example
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P(x) : x+1≥0
@@ -803,13 +803,13 @@ P(X) is True or False depending on the values of x

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Proprieties
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Proprieties
+

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Propriety Number 1:
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Propriety Number 1:
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The negation of the universal quantifier is the existential quantifier, and vice-versa :
@@ -820,8 +820,8 @@ The negation of the universal quantifier is the existential quantifier, and vice

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Example:
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Example:
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∀ x ≥ 1 x² > 5 ⇔ ∃ x ≥ 1 x² < 5
@@ -829,8 +829,8 @@ The negation of the universal quantifier is the existential quantifier, and vice

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Propriety Number 2:
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Propriety Number 2:
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∀x ∈ E, [P(x) ∧ Q(x)] ⇔ [∀ x ∈ E, P(x)] ∧ [∀ x ∈ E, Q(x)]
@@ -841,8 +841,8 @@ The propriety “For any value of x from a set E , P(x) and Q(x)” is e

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Example :
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Example :
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P(x) : sqrt(x) > 0 ; Q(x) : x ≥ 1
@@ -860,8 +860,8 @@ P(x) : sqrt(x) > 0 ; Q(x) : x ≥ 1

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Propriety Number 3:
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Propriety Number 3:
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∃ x ∈ E, [P(x) ∧ Q(x)] ⇒ [∃ x ∈ E, P(x)] ∧ [∃ x ∈ E, Q(x)]
@@ -872,8 +872,8 @@ P(x) : sqrt(x) > 0 ; Q(x) : x ≥ 1

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Example of why it’s NOT an equivalence :
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Example of why it’s NOT an equivalence :
+

P(x) : x > 5 ; Q(x) : x < 5
@@ -886,8 +886,8 @@ Of course there is no value of x such as its inferior and superior to 5 at the s

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Propriety Number 4:
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Propriety Number 4:
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[∀ x ∈ E, P(x)] ∨ [∀ x ∈ E, Q(x)] ⇒ ∀x ∈ E, [P(x) ∨ Q(x)]
@@ -901,16 +901,16 @@ Of course there is no value of x such as its inferior and superior to 5 at the s

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Multi-parameter proprieties :
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Multi-parameter proprieties :
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A propriety P can depend on two or more parameters, for convenience we call them x,y,z…etc

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Example :
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Example :
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P(x,y): x+y > 0
@@ -926,8 +926,8 @@ P(-2,-1) is a False one

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WARNING :
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WARNING :
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∀x ∈ E, ∃y ∈ F , P(x,y)
@@ -943,8 +943,8 @@ Are different because in the first one y depends on x, while in the second one,

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Example :
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Example :
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∀ x ∈ ℕ , ∃ y ∈ ℕ y > x -–— True
@@ -958,8 +958,8 @@ Are different because in the first one y depends on x, while in the second one,

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Proprieties :
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Proprieties :
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not(∀x ∈ E ,∃y ∈ F P(x,y)) ⇔ ∃x ∈ E, ∀y ∈ F not(P(x,y))

not(∃x ∈ E ,∀y ∈ F P(x,y)) ⇔ ∀x ∈ E, ∃y ∈ F not(P(x,y))
@@ -968,20 +968,20 @@ Are different because in the first one y depends on x, while in the second one,
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Methods of mathematical reasoning :
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Methods of mathematical reasoning :
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Direct reasoning :
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Direct reasoning :
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To show that an implication P ⇒ Q is true, we suppose that P is true and we show that Q is true

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Example:
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Example:
+

Let a,b be two Real numbers, we have to prove that a² + b² = 1 ⇒ |a + b| ≤ 2
@@ -1024,9 +1024,9 @@ a²+b²=1 ⇒ |a + b| ≤ 2 Which is what we wanted to prove, therefor the im

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Reasoning by the Absurd:
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Reasoning by the Absurd:
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To prove that a proposition is True, we suppose that it’s False and we must come to a contradiction
@@ -1037,8 +1037,8 @@ And to prove that an implication P ⇒ Q is true using the reasoning by the absu

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Example:
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Example:
+

Prove that this proposition is correct using the reasoning by the absurd : ∀x ∈ ℝ* , sqrt(1+x²) ≠ 1 + x²/2
@@ -1056,17 +1056,17 @@ sqrt(1+x²) = 1 + x²/2 ; 1 + x² = (1+x²/2)² ; 1 + x² = 1 + x^4/4 + x² ;

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Reasoning by contraposition:
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Reasoning by contraposition:
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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

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Reasoning by counter example:
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Reasoning by counter example:
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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
@@ -1074,20 +1074,20 @@ To prove that a proposition ∀x ∈ E, P(x) is false, all we have to do is find

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3eme Cours : Oct 9
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3eme Cours : Oct 9
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Reasoning by recurrence :
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Reasoning by recurrence :
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P is a propriety dependent of n ∈ ℕ. 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

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Example:
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Example:
+

Let’s prove that ∀ n ≥ 1 , (n,k=1)Σk = [n(n+1)]/2
@@ -1123,21 +1123,21 @@ For n ≥ 1. We assume that P(n) is true, OR : (n, k=1)Σk = n(n+1)/2. We

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4eme Cours : Chapitre 2 : Sets and Operations
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4eme Cours : Chapitre 2 : Sets and Operations
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Definition of a set :
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Definition of a set :
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A set is a collection of objects that share the sane propriety

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Belonging, inclusion, and equality :
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Belonging, inclusion, and equality :
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Let E be a set. If x is an element of E, we say that x belongs to E we write x ∈ E, and if it doesn’t, we write x ∉ E

A set E is included in a set F if all elements of E are elements of F and we write E ⊂ F ⇔ (∀x , x ∈ E ⇒ x ∈ F). We say that E is a subset of F, or a part of F. The negation of this propriety is : E ⊄ F ⇔ ∃x , x ∈ E and x ⊄ F
@@ -1146,13 +1146,13 @@ A set is a collection of objects that share the sane propriety

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Intersections and reunions :
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Intersections and reunions :
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Intersection:
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Intersection:
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E ∩ F = {x / x ∈ E AND x ∈ F} ; x ∈ E ∩ F ⇔ x ∈ F AND x ∈ F
@@ -1163,9 +1163,9 @@ x ∉ E ∩ F ⇔ x ∉ E OR x ∉ F

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Union:
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Union:
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E ∪ F = {x / x ∈ E OR x ∈ F} ; x ∈ E ∪ F ⇔ x ∈ F OR x ∈ F
@@ -1176,17 +1176,17 @@ x ∉ E ∪ F ⇔ x ∉ E AND x ∉ F

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Difference between two sets:
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Difference between two sets:
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E\F(Which is also written as : E - F) = {x / x ∈ E and x ∉ F}

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Complimentary set:
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Complimentary set:
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If F ⊂ E. E - F is the complimentary of F in E.
@@ -1197,52 +1197,52 @@ FCE = {x /x ∈ E AND x ∉ F} ONLY WHEN F IS A SUBSET OF E

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Symentrical difference
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Symentrical difference
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E Δ F = (E - F) ∪ (F - E) ; = (E ∪ F) - (E ∩ F)

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Proprieties :
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Proprieties :
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Let E,F and G be 3 sets. We have :

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Commutativity:
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Commutativity:
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E ∩ F = F ∩ E E ∪ F = F ∪ E

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Associativity:
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Associativity:
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E ∩ (F ∩ G) = (E ∩ F) ∩ G E ∪ (F ∪ G) = (E ∪ F) ∪ G

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Distributivity:
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Distributivity:
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E ∩ (F ∪ G) = (E ∩ F) ∪ (E ∩ G) E ∪ (F ∩ G) = (E ∪ F) ∩ (E ∪ G)

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Lois de Morgan:
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Lois de Morgan:
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If E ⊂ G and F ⊂ G ;
@@ -1252,33 +1252,33 @@ If E ⊂ G and F ⊂ G ;

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An other one:
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An other one:
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E - (F ∩ G) = (E-F) ∪ (E-G) ; E - (F ∪ G) = (E-F) ∩ (E-G)

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An other one:
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An other one:
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E ∩ ∅ = ∅ ; E ∪ ∅ = E

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And an other one:
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And an other one:
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E ∩ (F Δ G) = (E ∩ F) Δ (E ∩ G)

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And the last one:
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And the last one:
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E Δ ∅ = E ; E Δ E = ∅
@@ -1286,10 +1286,167 @@ E Δ ∅ = E ; E Δ E = ∅

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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} +
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Notes :
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+∅ ∈ 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 +
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Examples :
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+E = {a,b,c} // P(E)={∅, {a}, {b}, {c}, {a,b}, {b,c}, {a,c}, {a,b,c}} +
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Partition of a set :
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+We say that A is a partition of E if: +
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∀ x ∈ A , x ≠ 0
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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.
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The reunion of all elements of A is equal to E
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Cartesian products :
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+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 +
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Example :
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+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 +
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Some proprieties:
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ExF = ∅ ⇔ E=∅ OR F=∅
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ExF = FxE ⇔ E=F OR E=∅ OR F=∅
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E x (F∪G) = (ExF) ∪ (ExG)
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(E∪F) x G = (ExG) ∪ (FxG)
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(E∪F) ∩ (GxH) = (E ∩ G) x (F ∩ H)
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Generally speaking : (ExF) ∪ (GxH) ≠ (E∪G) x (F∪H)
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Binary relations in a set :
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Definition :
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+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 +
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Proprieties :
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+Let E be a set and R a relation defined in E +
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We say that R is reflexive if ∀ x ∈ E, xRx (for any element x in E,x is related to itself)
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We say that R is symmetrical if ∀ x,y ∈ E , xRy ⇒ yRx
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We say that R is transitive if ∀ x,y,z ∈ E (xRy , yRz) ⇒ xRz
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We say that R is anti-symmetrical if ∀ x,y ∈ E xRy AND yRx ⇒ x = y
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Equivalence relationship :
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+We say that R is a relation of equivalence in E if its reflexive, symetrical and transitive +
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Equivalence class :
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+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} +
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+
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The quotient set :
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+E/R = {̅a , a ∈ E} +
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Order relationship :
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+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. +
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The order R is called total if ∀ x,y ∈ E xRy OR yRx
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The order R is called partial if ∃ x,y ∈ E xR̅y AND yR̅x
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+
+
+
TODO Examples :
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+
+∀x,y ∈ ℝ , xRy ⇔ x²-y²=x-y +
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Prove that R is an equivalence relation
+
Let a ∈ ℝ, find ̅a
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+

Author: Crystal
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Created: 2023-10-13 Fri 16:58
+
Created: 2023-10-17 Tue 22:32

\ No newline at end of file -- cgit 1.4.1-2-gfad0

Algebra 1

Contenu de la Matiére

Contenu de la Matiére

Rappels et compléments (11H)

Rappels et compléments (11H)

Structures Algébriques (11H)

Structures Algébriques (11H)

Polynômes et fractions rationnelles

Polynômes et fractions rationnelles

Premier cours : Logique mathématique et méthodes du raisonnement mathématique Sep 25 :

Premier cours : Logique mathématique et méthodes du raisonnement mathématique Sep 25 :

Properties:

Properties:

Absorption:

Absorption:

Commutativity:

Commutativity:

Associativity:

Associativity:

Distributivity:

Distributivity:

Neutral element:

Neutral element:

Negation of a conjunction & a disjunction:

Negation of a conjunction & a disjunction:

Transitivity:

Transitivity:

Contraposition:

Contraposition:

God only knows what this property is called:

God only knows what this property is called:

Some exercices I found online :

Some exercices I found online :

USTHB 2022/2023 Section B :

USTHB 2022/2023 Section B :

2éme cours Oct 2

2éme cours Oct 2

Quantifiers

Quantifiers

Proprieties

Proprieties

Multi-parameter proprieties :

Multi-parameter proprieties :

Methods of mathematical reasoning :

Methods of mathematical reasoning :

Direct reasoning :

Direct reasoning :

Reasoning by the Absurd:

Reasoning by the Absurd:

Reasoning by contraposition:

Reasoning by contraposition:

Reasoning by counter example:

Reasoning by counter example:

3eme Cours : Oct 9

3eme Cours : Oct 9

Reasoning by recurrence :

Reasoning by recurrence :

4eme Cours : Chapitre 2 : Sets and Operations

4eme Cours : Chapitre 2 : Sets and Operations

Definition of a set :

Definition of a set :

Belonging, inclusion, and equality :

Belonging, inclusion, and equality :

Intersections and reunions :

Intersections and reunions :

Intersection:

Intersection:

Union:

Union:

Difference between two sets:

Difference between two sets:

Complimentary set:

Complimentary set:

Symentrical difference

Symentrical difference

Proprieties :

Proprieties :

Commutativity:

Commutativity:

Associativity: