Namespaces
Variants
Actions

Difference between revisions of "Injection"

From Encyclopedia of Mathematics
Jump to: navigation, search
m
(rewritten)
Line 1: Line 1:
''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" />''
 
  
A mapping $ f : A \to B $ 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" />.
+
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) = f(a_2) $, or equivalently $ f(a_1) = f(a_2) $ implies $ a_1 = a_2 $.
  
====Comments====
+
A special case is the ''inclusion'' function defined on a subset $ A \subset B $ by $ f(a)=a $.
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" />.
+
 
 +
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 [[Surjection|surjective]] is called [[Bijection|bijective]]  
 +
(or, if domain and range coincide, in some contexts, a [[permutation of a set|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''.

Revision as of 00:18, 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) = 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=21083
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article