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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 |