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.
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). (One-to-one function means one-to-one correspondence (i.e., bijection) to some authors, but injection to others.)
For example, consider the function succ, defined from the set of integers to
, 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.
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.
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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 o f of two bijections f X
Y and g
Y
Z is a bijection. The inverse of g o f is (g o f)−1 = (f −1) o (g−1).
On the other hand, if the composition g o f of two functions is bijective, we can only say that f is injective and g is surjective.
A relation f from X to Y is a bijective function if and only if there exists another relation g from Y to X such that g o f is the identity function on X, and f o g is the identity function on Y. Consequently, the sets have the same cardinality.
If X and Y are finite sets, then there exists a bijection between the two sets X and Y if and only if X and Y have the same number of elements. Indeed, in axiomatic set theory, this is taken as the very definition of "same number of elements", and generalising this definition to infinite sets leads to the concept of cardinal number, a way to distinguish the various sizes of infinite sets.
Formally, bijections are precisely the isomorphisms in the category Set of sets and functions. However, the bijections are not always the isomorphisms. For example, in the category Top of topological spaces and continuous functions, the isomorphisms must be homeomorphisms in addition to being bijections.