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Difference between revisions of "Injection"

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$ \def\Id {\mathop{\rm Id}} $
  
 
A function (or mapping) is called '''injective''' if distinct arguments have distinct images.
 
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
 
In other words, a function $ f : A \to B $ from a set $A$ to a set $B$ is
: an '''injective function''' or an '''injection'''  
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: an '''injective function''' or an '''injection''' or '''one-to-one''' function
 
if and only if
 
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 $
 
: $ 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 $.
 
for all $ a_1, a_2 \in A $.
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 +
==== Equivalent conditions ====
 +
 +
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 $. 
 +
(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 special case is the ''inclusion'' function defined on a subset $ A \subset B $ by $ f(a)=a $.
  
An injective [[homomorphism]] is called monomorphism.
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An injective [[homomorphism]] is called ''monomorphism''.
  
Injective mappings that are compatible with the underlying structure are often called [[embedding]].
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Injective mappings that are compatible with the underlying structure are often called ''[[embedding]]s''.
  
A function that is both injective and [[Surjection|surjective]] is called [[Bijection|bijective]]  
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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]]).
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(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 $. 
 
In category theory, this property is used to define ''monomorphisms''.
 

Revision as of 11:03, 19 February 2012

$ \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 $.

An injective homomorphism is called monomorphism.

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

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

How to Cite This Entry:
Injection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Injection&oldid=21203
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article