bijection
Let X and Y be sets. A function f:X→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→Y is a bijection, then for any A⊂X and B⊂Y we have
f∘f-1(B) | =B | ||
f-1∘f(A) | =A |
It easy to see the inverse of a bijection is a bijection, and that a composition 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 |