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 onetoone correspondence between those sets; i.e., both onetoone (injective) and onto (surjective).
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 X ↔ Y. (Sometimes this notation is reserved for binary relations, and bijections are denoted by X ⤖ Y 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.
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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 of two bijections and is a bijection. The inverse of is .
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