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Difference between revisions of "Bijection"

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(rewrite (similar to injective, surjective))
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''bijective mapping, of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016230/b0162301.png" /> into a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016230/b0162302.png" />''
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A function (or mapping) is called '''bijective'''
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if it is both one-to-one and onto, i.e.,
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if it is both [[Injection|injective]] and [[Surjection|surjective]].
  
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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016230/b0162303.png" /> under which different elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016230/b0162304.png" /> have different images in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016230/b0162305.png" />. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016230/b0162306.png" /> is a one-to-one mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016230/b0162307.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016230/b0162308.png" />, i.e. a mapping that is both an [[Injection|injection]] and a [[Surjection|surjection]]. A bijection establishes a [[One-to-one correspondence|one-to-one correspondence]] between the elements of the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016230/b0162309.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016230/b01623010.png" />. A bijection of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016230/b01623011.png" /> onto itself is also called a permutation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016230/b01623012.png" />.
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In other words, a function $ f : A \to B $ from a set $A$ to a set $B$ is
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: a '''bijective function''' or a '''bijection'''
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if and only if
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: $ f(A) = B $ and $ a_1 \ne a_2 $ implies $ f(a_1) \ne f(a_2) $ for all $ a_1, a_2 \in A $.
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In certain contexts, a bijective mapping of a (countable) set $A$ onto itself is called a [[permutation]] of $A$.
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A bijective [[homomorphism]] is called isomorphism.

Revision as of 01:17, 18 February 2012

A function (or mapping) is called bijective if it is both one-to-one and onto, i.e., if it is both injective and surjective.

In other words, a function $ f : A \to B $ from a set $A$ to a set $B$ is

a bijective function or a bijection

if and only if

$ f(A) = B $ and $ a_1 \ne a_2 $ implies $ f(a_1) \ne f(a_2) $ for all $ a_1, a_2 \in A $.

In certain contexts, a bijective mapping of a (countable) set $A$ onto itself is called a permutation of $A$.

A bijective homomorphism is called isomorphism.

How to Cite This Entry:
Bijection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bijection&oldid=21159
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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