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bijection (Definition)

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.



"bijection" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: function, permutation, injective function, surjective, isomorphism, uniqueness of cardinality, cardinality of disjoint union of finite sets, a connected normal space with more than one point is uncountable

Other names:  bijective, bijective function, 1-1 correspondence, 1 to 1 correspondence, one to one correspondence, one-to-one correspondence
Keywords:  Set

Attachments:
inverse function (Definition) by matte
example of bijection (Example) by juanman
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Cross-references: composition, easy to see, invertible, inverse function, consequence, permutation, onto, one-to-one, function
There are 198 references to this entry.

This is version 11 of bijection, born on 2001-10-20, modified 2007-05-12.
Object id is 425, canonical name is Bijection.
Accessed 37787 times total.

Classification:
AMS MSC: 03-00 (Mathematical logic and foundations :: General reference works )
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