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# bijection

Let $X$ and $Y$ be sets. A function $f\colon X\to Y$ that is one-to-one and onto is called a *bijection* or *bijective function* from $X$ to $Y$.

When $X=Y$, $f$ is also called a *permutation* of $X$.

An important consequence of the bijectivity of a function $f$ is the existence of an inverse function $f^{{-1}}$. Specifically, a function is invertible if and only if it is bijective. Thus if $f:X\rightarrow Y$ is a bijection, then for any $A\subset X$ and $B\subset Y$ we have

$\displaystyle f\circ f^{{-1}}(B)$ | $\displaystyle=B$ | ||

$\displaystyle f^{{-1}}\circ f(A)$ | $\displaystyle=A$ |

It easy to see the inverse of a bijection is a bijection, and that a composition of bijections is again bijective.

Keywords:

Set

Related:

Function, Permutation, InjectiveFunction, Surjective,Isomorphism2, CardinalityOfAFiniteSetIsUnique, CardinalityOfDisjointUnionOfFiniteSets, AConnectedNormalSpaceWithMoreThanOnePointIsUncountable2, AConnectedNormalSpaceWithMoreThanOnePointIsUncountable, Bo

Synonym:

bijective, bijective function, 1-1 correspondence, 1 to 1 correspondence, one to one correspondence, one-to-one correspondence

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

03-00*no label found*83-00

*no label found*81-00

*no label found*82-00

*no label found*

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