Let X and Y be sets. A function f:XY 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 permutationMathworldPlanetmath 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 invertiblePlanetmathPlanetmath if and only if it is bijectiveMathworldPlanetmath. Thus if f:XY is a bijection, then for any AX and BY we have

ff-1(B) =B
f-1f(A) =A

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

Title bijection
Canonical name Bijection
Date of creation 2013-03-22 11:51:35
Last modified on 2013-03-22 11:51:35
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 16
Author mathcam (2727)
Entry type Definition
Classification msc 03-00
Classification msc 83-00
Classification msc 81-00
Classification msc 82-00
Synonym bijective
Synonym bijective function
Synonym 1-1 correspondence
Synonym 1 to 1 correspondence
Synonym one to one correspondence
Synonym one-to-one correspondence
Related topic Function
Related topic Permutation
Related topic InjectiveFunction
Related topic Surjective
Related topic Isomorphism2
Related topic CardinalityOfAFiniteSetIsUnique
Related topic CardinalityOfDisjointUnionOfFiniteSets
Related topic AConnectedNormalSpaceWithMoreThanOnePointIsUncountable2
Related topic AConnectedNormalSpaceWithMoreThanOnePointIsUncountable
Related topic Bo