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 and 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). (Onetoone function means onetoone 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 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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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 .
On the other hand, if the composition 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 is the identity function on X, and 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 iff 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.
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A bijection (or bijective function) is a function that is both injective and surjective. A bijection is sometimes called a onetoone mapping, or a onetoone correspondence.
This is the same as saying: a function $f$ with domain $A$ and codomain $B$ is bijective if and only if $f(x)$ and $f(y)$ are different whenever $x$ and $y$ are different, and every element $z$ of $B$ has an element $x$ of $A$ where $f(x)=z$.
If $f$ is a bijection from $A$ to $B$ then its inverse, $f^\{1\}$, is a bijection from $B$ to $A$.
