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

f: A -> BThis means that the function

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*.

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