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Difference between revisions of "Injection"

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: an '''injective function''' or an '''injection'''  
 
: an '''injective function''' or an '''injection'''  
 
if and only if
 
if and only if
: $ a_1 \not= a_1 \in A $ implies $ f(a_1) = f(a_2) $, or equivalently $ f(a_1) = f(a_2) $ implies $ a_1 = a_2 $.
+
: $ a_1 \not= a_1 \in A $ implies $ f(a_1) \ne f(a_2) $, or equivalently $ f(a_1) = f(a_2) $ implies $ a_1 = a_2 $.
  
 
A special case is the ''inclusion'' function defined on a subset $ A \subset B $ by $ f(a)=a $.
 
A special case is the ''inclusion'' function defined on a subset $ A \subset B $ by $ f(a)=a $.

Revision as of 07:11, 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 \not= a_1 \in A $ implies $ f(a_1) \ne f(a_2) $, or equivalently $ f(a_1) = f(a_2) $ implies $ a_1 = a_2 $.

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 $ f \circ g = f \circ h $ then $ g=h $. In category theory, this property is used to define injective morphisms.

How to Cite This Entry:
Injection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Injection&oldid=21087
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article