4. function

function

A function f from a set X to a set Y(f : X->Y) is a subset of XxY such that for all x∈X, there is exactly one y∈Y with (x, y)∈f.

floor and ceiling

The floor of x, denoted └x┘, is the greatest integer less than or equal to x.

The ceiling of x, denoted ┌x┐, is the least integer greater than or equal to x.

Properties of a function f 

- one-to-one                          for all x1, x2∈X, if(x1) = f(x2), then x1 = x2.

- onto                                   for all y∈Y, there exists x∈X such that y=f(x).

- bijection                              one-to-one, onto

- Binary operator on a set X       a function from X x X to X

- unary operator on X               a function from X to X

inverse of f

Let f be a bijection function from X to Y.

the inverse of f, denoted f^-1, is the funtion from Y to X defined by

f^-1 = { (y, x)| (x, y) ∈ f}

composition

Let g be a function from X to Y and f be a funtion from Y to Z.

the composition of f with g, denoted f ∘ g, is the funtion from X to Z defined by

f ∘g(x) = f( g(x) )