Prove that `f:X rarr Y` is injective iff for all `subsets A,B of X,f(A nn B ) = f(A) nn f(B)`.

Prove that f:X to Y is injective iff for all subsets A, B of X, f (A cap B) =f(A) cap f(B) .

Prove that f:X rarr Y is surjective iff for all B sube Y, f(f^(-1)(B)) =B.

Prove that f:X rarr Y is injective iff f^(-1) (f(A)) = "A for all" A sube X .

Prove that f: X rarr Y is surjective iff for all A sube X,(f(A))' sube f(A') , where A' denotes the complement of A in X.

Prove that for any f:X rarr Y , f o id_x = f =id_Y of.

Prove that A nn(B uu C) =(A nn B) uu (A nn C)

Prove that f(x)=|x| is not differetiable at x=0

Using the definition, prove that the function f: A rarr B is invertible, if and only if f is both one-one and onto.

Let f be defined by f (x)=2x + 1 for all xinR . Show that f is bijective and determine the inverse of f. Find f^(-1)(0),f^(-1)(2) and f^(-1)(-1) .