# Bijection

 related topics {math, number, function}

In mathematics, a bijection, or a bijective function, is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that f(x) = y. It follows from this definition that no unmapped element exists in either X or Y.

Alternatively, f is bijective if it is a one-to-one correspondence between those sets; i.e., both one-to-one (injective) and onto (surjective).

For example, consider the function succ, defined from the set of integers $\Z$ to $\Z$, that to each integer x associates the integer succ(x) = x + 1. For another example, consider the function sumdif that to each pair (x,y) of real numbers associates the pair sumdif(x,y) = (x + y, x − y).

A bijective function from a set to itself is also called a permutation.

The set of all bijections from X to Y is denoted as XY. (Sometimes this notation is reserved for binary relations, and bijections are denoted by XY instead.) Occasionally, the set of permutations of a single set X may be denoted X!.

Bijective functions play a fundamental role in many areas of mathematics, for instance in the definition of isomorphism (and related concepts such as homeomorphism and diffeomorphism), permutation group, projective map, and many others.

## Contents

### Composition and inverses

A function f is bijective if and only if its inverse relation f −1 is a function. In that case, f −1 is also a bijection.

The composition $g\circ f$ of two bijections $f:X\to Y$ and $g:Y\to Z$ is a bijection. The inverse of $g\circ f$ is $(g\circ f)^{-1}=(f^{-1})\circ(g^{-1})$.