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+++ b/src/org/uni_notes/algebra1.org
@@ -472,7 +472,7 @@ If F ⊂ E. E - F is the complimentary of F in E.
 
 
 FCE = {x /x ∈ E AND x ∉ F} *ONLY WHEN F IS A SUBSET OF E*
-*** Symentrical difference
+*** Symmetrical difference
 E Δ F = (E - F) ∪ (F - E) ; = (E ∪ F) - (E ∩ F)
 ** Proprieties :
 Let E,F and G be 3 sets. We have :
@@ -594,3 +594,101 @@ De (1) et (2), P(0) est vraie ---- (a)
 Donc P(n+1) est vraie. ---- (b)
 De (a) et (b) on conclus que la proposition de départ est vraie
 #+END_VERSE
+* Chapter 3 : Applications
+** 3.1 Generalities about applications :
+*** Definition :
+Let E and F be two sets.
+1. We call a function of the set E to the set F any relation from E to F such as for any element of E, we can find _at most one_ element of F that corresponds to it.
+2. We call an application of the set E to the set F a relation from E to F such as for any element of E, we can find _one and only one_ element of F that corresponds to it.
+3. f: E_{1} ---> F_{1} ; g: E_{2} ---> F_{2} ; f ≡ g ⇔ [E_{1 }= E_{2} ; F_{1} = F_{2} ; f(x) = g(x) ∀x ∈ E_{1}
+
+   Generally speaking, we schematize a function or an application by this writing :
+   #+BEGIN_VERSE
+   f : E ---> F
+       x ---> f(x)=y
+      Γ = {(x , f(x))/ x ∈ E ; f(x) ∈ F} is the graph of f
+   #+END_VERSE
+**** Some examples :
+***** Ex1:
+#+BEGIN_VERSE
+f : ℝ ---> ℝ
+    x ---> f(x) = (x-1)/x
+is a function, because 0 does NOT have a corresponding element using that relation.
+#+END_VERSE
+***** Ex2:
+#+BEGIN_VERSE
+f : ℝ^{*} ---> ℝ
+    x ---> f(x)= (x-1)/x
+is, however, an application
+#+END_VERSE
+*** Restriction and prolongation of an application :
+Let f : E -> F an application and E_{1} ⊂ E therefore :
+#+BEGIN_VERSE
+g : E_{1} -> F
+g(x) = f(x) ∀x ∈ E_{1}
+
+g is called the *restriction* of f to E_{1}. And f is called the *prolongation* of g to E.
+#+END_VERSE
+**** Example
+#+BEGIN_VERSE
+f : ℝ ---> ℝ
+    x ---> f(x) = x^{2}
+
+g : [0 , +∞[ ---> ℝ
+    x ---> g(x) = x²
+
+g is called the *restriction* of f to ℝ^{+}. And f is called the *prolongation* of g to ℝ.
+#+END_VERSE
+*** Composition of applications :
+Let E,F, and G be three sets, f: E -> F and g: F -> G are two applications. We define their composition, symbolized by g_{o}f as follow :
+
+
+g_{o}f : E -> G . ∀x ∈ E (g_{o}f)_{(x)}= g(f(x))
+** 3.2 Injection, surjection and bijection :
+Let f: E -> F be an application :
+1. We say that f is injective if : ∀x,x' ∈ E : f(x) = f(x') ⇒ x = x'
+2. We say that f is surjective if : ∀ y ∈ F , ∃ x ∈ E : y = f(x)
+3. We say that if is bijective if it's both injective and surjective at the same time.
+*** Proposition :
+Let f : E -> F be an application. Therefore:
+1. f is injective ⇔ y = f(x) has at most one solution.
+2. f is surjective ⇔ y = f(x) has at least one solution.
+3. f is bijective ⇔ y = f(x) has a single and unique solution.
+** 3.3 Reciprocal applications :
+*** Def :
+Let f : E -> F a bijective application. So there exists an application named f^{-1} : F -> E such as : y = f(x) ⇔ x = f^{-1}(y)
+*** Theorem :
+Let f : E -> F be a bijective application. Therefore its reciprocal f^{-1} verifies : f^{-1}_{o}f=Id_{E }; f_{o}f^{-1}=Id_{F} Or :
+
+
+Id_{E} : E -> E ; x -> Id_{E}(x) = x
+*** Some proprieties :
+1. (f^{-1})^{-1} = f
+2. (g_{o}f)⁻¹ = f⁻¹_{o}g⁻¹
+3. The graphs of f and f⁻¹ are symmetrical to each other by the first bis-sectrice of the equation y = x
+** 3.4 Direct Image and reciprocal Image :
+*** Direct Image :
+   Let f: E-> F be an application and A ⊂ E. We call a direct image of A by f, and we symbolize as f(A) the subset of F defined by :
+
+
+f(A) = {f(x)/ x ∈ A} ; = { y ∈ F ∃ x ∈ A  y=f(x)}
+**** Example :
+#+BEGIN_VERSE
+f: ℝ -> ℝ
+   x -> f(x) = x²
+A = {0,4}
+f(A) = {f(0), f(4)} = {0, 16}
+#+END_VERSE
+*** Reciprocal image :
+Let f: E -> F be an application and B ⊂ F. We call the reciprocal image of E by F the subset f^{-1}(B) :
+
+
+f^{-1}(B) = {x ∈ E/f(x) ∈ B} ; x ∈ f^{-1}(B) ⇔ f(x) ∈ B
+**** Example :
+#+BEGIN_VERSE
+f: ℝ -> ℝ
+   x -> f(x) = x²
+B = {1,9,4}
+f^{-1}(B) = {1,-1,2,-2,3,-3}
+      = {x ∈ ℝ/x² ∈ {1,4,9}}
+#+END_VERSE