From 16e5f6e90b603bc62d4c0af24d79ca5e6121ac6a Mon Sep 17 00:00:00 2001
From: Crystal <crystal@wizard.tower>
Date: Tue, 17 Oct 2023 22:42:30 +0100
Subject: Finally, an update

---
 src/org/uni_notes/algebra1.org | 58 ++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 58 insertions(+)

(limited to 'src')

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
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