application injective et surjective pdf

Nov 10 2019 · Injective Functions Formal Defintion: A function f : D ! C is injective if and only if “for all x1 2 D and x2 2 D if f(x1) = f(x2) then x1 = x2 ” Casual Definition: No two distinct points in the domain map to the same value Classic Example: f(x) = ex thought of as f : R ! R Horizontal Line Test: Every horizontal line hits the curve at most once

Exemple : La fonction carrée est surjective de R sur R+ 3 2 Surjectivité et composition Théorème 2 : Soient f et g deux applications f: E → F et g: F → G • Si f et g sont surjectives alors g f est surjective • Si g f est surjective alors g est surjective Remarque : Si g f est surjective alors f rien du tout Contre-exemple f: (R

f est surjective si et seulement si Im f ==fE()F 3 3 Bijectivité Proposition 3 Soit f ∈L(EF) f est dite bijective si et seulement si elle est à la fois injective et surjective • Tout élément de E possède une image unique dans F • Tout élément de F possède un antécédent unique dans E Théorème de la dimension

A \\ surjection\" is a surjective function Example Suppose that S = f1; 2; 3; 4g and T = fa; b; cg Then the map f : S ! T de ned by f(1) = a f(2) = c f(3) = b f(4) = a is surjective The function g : S ! T de ned by g(1) = a g(2) = b g(3) = a g(4) = b is not surjective since g doesn\'t send anything to c

Worksheet 15: Review functions: injective surjec-tive bijective functions Range 1 Determine the range of the functions f : R !R de ned as follows: (a) f(x) = x2 1 + x2 (b) f(x) = x 1 + jxj Solution a) f(x) = x2 1 + x2 Claim: f(R) = [0;1) Proof: ( ) For any real number r 2R we have that 0 r2 < 1 + r2

De nition A function f : X ! Y is surjective (also called onto) if every element y 2 Y is in the image of f that is if for any y 2 Y there is some x 2 X with f(x) = y Example The example f(x) = x2 as a function from R ! R is also not onto as negative numbers aren\'t squares of real numbers