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Difference between revisions of "Injection"

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("a1 ne a2 in A" is too informal)
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(or, if domain and range coincide, in some contexts, a [[permutation of a set|permutation]]).
 
(or, if domain and range coincide, in some contexts, a [[permutation of a set|permutation]]).
 
   
 
   
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 $.   
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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''.
 
In category theory, this property is used to define ''monomorphisms''.

Revision as of 13:34, 16 February 2012

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

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 $.

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

An injective homomorphism is called monomorphism.

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

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

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.

How to Cite This Entry:
Injection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Injection&oldid=21094
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article