Relations and Functions — Formula Sheet

R ⊆ A × A

R = ∅

R = A × A

(a,a) ∈ R for every a ∈ A

(a1,a2) ∈ R ⇒ (a2,a1) ∈ R

f(x1)=f(x2) ⇒ x1=x2

For every y in Y, there exists x in X such that f(x)=y

One-one and onto

f(x)=2x, f:N→N

f(x)=2x, f:R→R

gof(x) = g(f(x)), ∀x∈A

A function f:X→Y is invertible iff there exists g:Y→X such that gof=I_X and fog=I_Y

Invertible ⇔ one-one and onto

f(x)=4x+3; g(y)=(y-3)/4

f(x)=x/(1+|x|)