A function is defined as a mapping between two sets. The notation for this is

 f: A -> B
This means that the function ''f'' maps an element from set ''A'' to an element in set ''B''. Or, more informally, ''f'' takes an argument of type ''A'', and returns an argument of type ''B''.

Functions come in 3 flavours: Injection, Surjection and Bijection.

'''Injection'''

A function f: A -> B is ''one-to-one'' if for all a1, a2 in A, a1 != a2 implies f(a1) != f(a2).

This is also known as an injection. Informally, it means that no two elements in A map to the same element in B.

'''Surjection'''

A function g: A -> B is ''onto'' if for all b in B, there exists a in A with g(a) = b.

This is also known as a surjection. Informally, it means that all elements in B are mapped to. It is possible, unlike an injection, to have an element of B mapped to by more than one element in A.

'''Bijection'''

A function is a bijection if it is both an injection and a surjection.

If f: A -> B is one-to-one and onto, then it provides a ''one-to-one correspondence'' between the sets A and B. Informally, every element of A maps to exactly one element of B, and all elements of B are mapped to.

You often see ''one-to-one'' (mis)used to mean ''there is a one-to-one correspondence''.