# 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.

 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