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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
It easy to see the inverse of a bijection is a bijection, and that a composition of bijections is again bijective.
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"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, Borel isomorphism
| Other names: |
bijective, bijective function, 1-1 correspondence, 1 to 1 correspondence, one to one correspondence, one-to-one correspondence |
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Cross-references: composition, easy to see, invertible, inverse function, consequence, permutation, onto, one-to-one, function
There are 236 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 49204 times total.
Classification:
| AMS MSC: | 03-00 (Mathematical logic and foundations :: General reference works ) |
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Pending Errata and Addenda
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