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''injective mapping, of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051200/i0512001.png" /> into a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051200/i0512002.png" />''
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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051200/i0512003.png" /> under which different elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051200/i0512004.png" /> have different images in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051200/i0512005.png" />. An injection is also called an imbedding (or inclusion) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051200/i0512006.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051200/i0512007.png" />.
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$ \def\Id {\mathop{\rm Id}} $
  
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A function (or mapping) is called '''injective''' if distinct arguments have distinct images.
  
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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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: an '''injective function''' or an '''injection''' or '''one-to-one''' function
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if and only if
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: $ a_1 \ne a_2 $ implies $ f(a_1) \ne f(a_2) $, or equivalently $ f(a_1) = f(a_2) $ implies $ a_1 = a_2 $
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for all $ a_1, a_2 \in A $.
  
====Comments====
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==== Equivalent conditions ====
In [[Category|category]] theory, a [[Morphism|morphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051200/i0512008.png" /> is called injective if for all morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051200/i0512009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051200/i05120010.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051200/i05120011.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051200/i05120012.png" />.
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A function $f$ is injective if and only if
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$ f^{-1}(f(S)) = S $ for all subsets $S$ of the domain $A$.
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A function $f$ is injective if and only if, for every pair  of functions $g,h$ with values in $A$,
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the condition $ f \circ g = f  \circ h $ implies $ g=h $.
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(In category theory, this property is used to define ''monomorphisms''.)
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A function $f$ is injective if and only if
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there is a left-inverse function $g$ with $ g \circ f = \Id_A$.
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==== Related notions ====
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A special case is the ''inclusion'' function defined on a subset $ A \subset B $ by $ f(a)=a $.
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A function that is both injective and [[Surjection|surjective]] is called ''[[Bijection|bijective]]''
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(or, if domain and range coincide, in some contexts, a ''[[permutation of a set|permutation]])''.
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An injective [[homomorphism]] is called ''monomorphism''.
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Injective mappings that are compatible with the underlying structure are often called ''[[embedding]]s''.

Latest revision as of 12:11, 12 December 2013


$ \def\Id {\mathop{\rm Id}} $

A function (or mapping) is called injective if distinct arguments have distinct images.

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

an injective function or an injection or one-to-one function

if and only if

$ a_1 \ne a_2 $ implies $ f(a_1) \ne f(a_2) $, or equivalently $ f(a_1) = f(a_2) $ implies $ a_1 = a_2 $

for all $ a_1, a_2 \in A $.

Equivalent conditions

A function $f$ is injective if and only if $ f^{-1}(f(S)) = S $ for all subsets $S$ of the domain $A$.

A function $f$ is injective if and only if, for every pair of functions $g,h$ with values in $A$, the condition $ f \circ g = f \circ h $ implies $ g=h $. (In category theory, this property is used to define monomorphisms.)

A function $f$ is injective if and only if there is a left-inverse function $g$ with $ g \circ f = \Id_A$.

Related notions

A special case is the inclusion function defined on a subset $ A \subset B $ by $ f(a)=a $.

A function that is both injective and surjective is called bijective (or, if domain and range coincide, in some contexts, a permutation).

An injective homomorphism is called monomorphism.

Injective mappings that are compatible with the underlying structure are often called embeddings.

How to Cite This Entry:
Injection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Injection&oldid=11558
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article