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''single-valued''
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{{TEX|done}}
  
A law according to which to every element of a given set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m0622701.png" /> has been assigned a completely defined element of another given set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m0622702.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m0622703.png" /> may coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m0622704.png" />). Such a relation between the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m0622705.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m0622706.png" /> is denoted in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m0622707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m0622708.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m0622709.png" />. One also writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227010.png" /> and says that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227011.png" /> operates from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227012.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227013.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227014.png" /> is called the domain (of definition) of the mapping, while the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227015.png" /> is called the range (of values) of the mapping. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227016.png" /> is also called a mapping of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227017.png" /> into the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227018.png" /> (or onto the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227019.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227020.png" />). Logically, the concept of a  "mapping"  coincides with the concept of a [[Function|function]], an [[Operator|operator]] or a [[Transformation|transformation]].
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$
 +
  \def\P{\mathcal P}        % power set
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  \def\iff{\Leftrightarrow}
 +
$
  
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227021.png" /> gives rise to a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227022.png" />, which is called the graph of the mapping. On the other hand, a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227023.png" /> defines a single-valued mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227024.png" /> having graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227025.png" /> if and only if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227026.png" /> one and only one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227027.png" /> exists such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227028.png" />; and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227029.png" />.
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'''Mapping''', or abbreviated '''map''', is one of many synonyms used for [[function]].
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In particular, the term map(ping) is used in general contexts, such as set theory, but usage is not restricted to these cases.
  
Two mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227031.png" /> are said to be equal if their domains of definition coincide and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227032.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227033.png" />. In this case the ranges of these mappings also coincide. A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227034.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227035.png" /> is constant if there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227036.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227037.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227038.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227039.png" /> defined on a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227040.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227041.png" /> by the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227043.png" />, is called the restriction of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227044.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227045.png" />; this restriction is often denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227046.png" />. A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227047.png" /> defined on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227048.png" /> and satisfying the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227049.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227050.png" /> is called an extension (or continuation) of the mapping to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227051.png" />. If three sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227052.png" /> are given, if a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227053.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227054.png" /> is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227055.png" />, and a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227056.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227057.png" /> is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227058.png" />, then there exists a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227059.png" /> with domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227060.png" />, taking values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227061.png" />, and defined by the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227062.png" />. This mapping is called the composite of the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227064.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227066.png" /> are called component (factor) mappings. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227067.png" /> is also called the compound mapping (composite mapping, composed mapping), consisting of the interior mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227068.png" /> and the exterior mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227069.png" />. The composed mapping is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227070.png" />, where the order of the notation is vital (for functions of a real variable, the term superposition is also used). The concept of a compound mapping can be generalized to any finite number of components of the mapping.
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==== The mapping concept in set theory ====
  
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227071.png" />, defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227072.png" /> and taking values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227073.png" />, gives rise to a new mapping defined on the subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227074.png" /> and taking subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227075.png" /> as values. In fact, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227076.png" />, then
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In [[set theory]] mappings are special [[binary relation]]s.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227077.png" /></td> </tr></table>
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A mapping $f$ from a set $A$ to a set $B$ is
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an (ordered) triple $ f = (A,B,G_f) $ where $ G_f \subset A \times B $
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such that
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* (a) if $ (x,y) $ and $ (x,y') \in G_f $ then $ y=y' $, and
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* (b) the projection $ \pi_1 (G_f) = \{ x \mid (x,y) \in G_f \} = A $.
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Condition (a) expresses that $f$ is ''single-valued''. and
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condition (b) that it is ''defined on'' $A$.
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<br>
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$A$ is the ''domain'', $B$ is the ''codomain'', and $G_f$ is the ''graph'' of the mapping.
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Therefore, in this setting, mappings are ''equal'' if and only if
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all three corresponding components (domain, codomain, and graph) are equal.
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<br>
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The mapping is usually denoted as $ f : A \to B $, and $ a \mapsto f(a) $
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where $ f(a) := b \iff (a,b) \in G_f $ is the ''value'' of $f$ at $a$.
  
The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227078.png" /> is called the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227079.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227080.png" />, the initial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227081.png" /> is obtained; thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227082.png" /> is an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227083.png" /> from the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227084.png" /> to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227085.png" /> of all subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227086.png" /> if a one-element set is identified with the element comprising it. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227087.png" />, a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227088.png" /> is called an invariant subset for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227089.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227090.png" />, while a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227091.png" /> is called a fixed point for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227092.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227093.png" />. Invariant sets and fixed points are important in solving functional equations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227094.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227095.png" />.
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If two mappings $ f_1 = (A_1,B_1,G_1) $ and $ f_2 = (A_2,B_2,G_2) $ satisfy
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: $ A_1 \subset A_2 $, $ B_1 \subset B_2 $ and $ G_1 \subset G_2 $
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then $f_2$ is called an ''extension'' of $ f_1 $, and $ f_1 $ a ''restriction'' of $f_2$.
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In this case, $ f_1 $ is often denoted as $ f_2 \vert A_1 $
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and, clearly, $ f_1 (a) = f_2 (a) $ holds for all $ a \in A_1 $.
  
Every mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227096.png" /> gives rise to a mapping defined on the subsets of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227097.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227098.png" /> and taking subsets of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m06227099.png" /> as values. In fact, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270100.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270101.png" />), the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270102.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270103.png" />, and is called the complete inverse image (complete pre-image) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270104.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270105.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270106.png" /> consists of a single element, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270107.png" /> is a mapping of elements, is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270108.png" />, and takes values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270109.png" />. It is also called the inverse mapping for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270110.png" />. The existence of an inverse mapping is equivalent to the solvability of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270111.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270112.png" />, for a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270113.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270114.png" /> is given.
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''Remark:''<br>
 +
Sometimes only the graph $G_f$ is used to represent a function.
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In this case two mappings are equal if they have the same graph,
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and one may allow graphs that are not sets but classes.
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<br>
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While the domain of the function can be obtained as projection $ \pi_1 (G_f) $ of the first component,  
 +
the projection $ \pi_2 (G_f) $ of the second component does not produce the codomain but only the image of the domain.
 +
Thus the concept of [[surjection|surjectivity]] is not applicable.
  
If the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270115.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270116.png" /> have certain properties, then interesting classes can be distinguished in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270117.png" /> of all mappings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270118.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270119.png" />. Thus, for partially ordered sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270120.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270121.png" />, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270122.png" /> is isotone if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270123.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270124.png" /> (cf. [[Isotone mapping|Isotone mapping]]). For complex planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270125.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270126.png" />, the class of holomorphic mappings is naturally selected. For topological spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270127.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270128.png" />, the class of continuous mappings between these spaces is distinguished naturally; an extended theory of differentiation of mappings (cf. [[Differentiation of a mapping|Differentiation of a mapping]]) has been constructed. For mappings of a scalar argument and, in the most general case, for mappings defined on a [[Measure space|measure space]], the concept of (weak or strong) measurability can be introduced, and various Lebesgue-type integrals can be constructed (for example, the [[Bochner integral|Bochner integral]] and the [[Daniell integral|Daniell integral]]).
+
==== Composition ====
  
A mapping is called a multi-valued mapping if subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270129.png" /> consisting of more than one element are assigned to certain values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270130.png" />. Examples of this type of mappings include multi-sheeted functions of a complex variable, multi-valued mappings of topological spaces, and others.
+
Two mappings can be ''composed'' if the codomain of one mapping is a subset of the domain of the other mapping:
  
====References====
+
For $ f=(A,B,G_f) $ and $ g=(C,D,G_g) $ with $ B \subset C $
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Theory of sets" , Addison-Wesley  (1968) (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. General topology" , Addison-Wesley  (1966) (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.L. Kelley,   "General topology" , Springer  (1975)</TD></TR></table>
+
the ''composition'' $ g \circ f $ is the mapping $ (A,D,G) $ with
 +
: $ G := \{ (a,g(f(a))) \mid a \in A \} = \{ (a,c) \mid (\exists b \in B) ( (a,b) \in G_f \land (b,c) \in G_g ) \} $.
  
 +
''Remarks:'' <br>
 +
(a) The condition $ B \subset C $ can be relaxed to $ f(A) \subset C $.
 +
<br>
 +
(b) If only graphs are used then the graph of the composition is defined (as above) by
 +
: $ G_{g \circ f} := \{ (a,c) \mid (\exists b ) ( (a,b) \in G_f \land (b,c) \in G_g ) \} $
 +
but may turn out to be empty.
  
 +
==== Induced mappings ====
  
====Comments====
+
Every mapping $ f : A \to B $ induces two mappings between the [[power set]]s $\P(A)$ and $\P(B)$.
For a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270131.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270132.png" /> is also called the source of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270133.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270134.png" /> is also called the target of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062270/m062270135.png" />, [[#References|[a3]]].
+
: $ f_\ast : \P(A) \to \P(B) $ defined by $ f_\ast (S) := \{ f(a) \mid a \in S \}$ for $ S \subset A $
 +
and
 +
: $ f^\ast : \P(B) \to \P(A) $ defined by $ f^\ast (T) := \{ a \mid f(a) \in T \}$ for $ T \subset B $
 +
$ f_\ast (S) $ is called the ''image'' of $S$ under $f$, usually denoted as $f(S)$, and
 +
$ f^\ast (T) $ is called the ''inverse image'' of $T$ under $f$, usually denoted as $f^{-1}(T)$,
 +
but one has to be aware that these common notations may be ambiguous in certain situations.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.R. Halmos,  "Measure theory" , v. Nostrand (1950)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.R. Halmos,   "Naive set theory" , v. Nostrand  (1961)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P.M. Cohn,   "Universal algebra" , Reidel  (1981)</TD></TR></table>
+
N. Bourbaki,  "Elements of mathematics. Theory of sets" , Addison-Wesley (1968(Translated from French)
 +
 
 +
Paul R. Halmos, ''Naive Set Theory.''
 +
<br>&nbsp;&nbsp; (The University Series in Undergraduate Mathematics) Princeton, N. J., etc., Van Nostrand, 1960.
 +
<br>&nbsp;&nbsp; ''Reprinted'': (Undergraduate Texts in Mathematics) New York, etc., Springer, 1974.

Latest revision as of 11:54, 20 March 2016


$ \def\P{\mathcal P} % power set \def\iff{\Leftrightarrow} $

Mapping, or abbreviated map, is one of many synonyms used for function. In particular, the term map(ping) is used in general contexts, such as set theory, but usage is not restricted to these cases.

The mapping concept in set theory

In set theory mappings are special binary relations.

A mapping $f$ from a set $A$ to a set $B$ is an (ordered) triple $ f = (A,B,G_f) $ where $ G_f \subset A \times B $ such that

  • (a) if $ (x,y) $ and $ (x,y') \in G_f $ then $ y=y' $, and
  • (b) the projection $ \pi_1 (G_f) = \{ x \mid (x,y) \in G_f \} = A $.

Condition (a) expresses that $f$ is single-valued. and condition (b) that it is defined on $A$.

$A$ is the domain, $B$ is the codomain, and $G_f$ is the graph of the mapping. Therefore, in this setting, mappings are equal if and only if all three corresponding components (domain, codomain, and graph) are equal.
The mapping is usually denoted as $ f : A \to B $, and $ a \mapsto f(a) $ where $ f(a) := b \iff (a,b) \in G_f $ is the value of $f$ at $a$.

If two mappings $ f_1 = (A_1,B_1,G_1) $ and $ f_2 = (A_2,B_2,G_2) $ satisfy

$ A_1 \subset A_2 $, $ B_1 \subset B_2 $ and $ G_1 \subset G_2 $

then $f_2$ is called an extension of $ f_1 $, and $ f_1 $ a restriction of $f_2$. In this case, $ f_1 $ is often denoted as $ f_2 \vert A_1 $ and, clearly, $ f_1 (a) = f_2 (a) $ holds for all $ a \in A_1 $.

Remark:
Sometimes only the graph $G_f$ is used to represent a function. In this case two mappings are equal if they have the same graph, and one may allow graphs that are not sets but classes.
While the domain of the function can be obtained as projection $ \pi_1 (G_f) $ of the first component, the projection $ \pi_2 (G_f) $ of the second component does not produce the codomain but only the image of the domain. Thus the concept of surjectivity is not applicable.

Composition

Two mappings can be composed if the codomain of one mapping is a subset of the domain of the other mapping:

For $ f=(A,B,G_f) $ and $ g=(C,D,G_g) $ with $ B \subset C $ the composition $ g \circ f $ is the mapping $ (A,D,G) $ with

$ G := \{ (a,g(f(a))) \mid a \in A \} = \{ (a,c) \mid (\exists b \in B) ( (a,b) \in G_f \land (b,c) \in G_g ) \} $.

Remarks:
(a) The condition $ B \subset C $ can be relaxed to $ f(A) \subset C $.
(b) If only graphs are used then the graph of the composition is defined (as above) by

$ G_{g \circ f} := \{ (a,c) \mid (\exists b ) ( (a,b) \in G_f \land (b,c) \in G_g ) \} $

but may turn out to be empty.

Induced mappings

Every mapping $ f : A \to B $ induces two mappings between the power sets $\P(A)$ and $\P(B)$.

$ f_\ast : \P(A) \to \P(B) $ defined by $ f_\ast (S) := \{ f(a) \mid a \in S \}$ for $ S \subset A $

and

$ f^\ast : \P(B) \to \P(A) $ defined by $ f^\ast (T) := \{ a \mid f(a) \in T \}$ for $ T \subset B $

$ f_\ast (S) $ is called the image of $S$ under $f$, usually denoted as $f(S)$, and $ f^\ast (T) $ is called the inverse image of $T$ under $f$, usually denoted as $f^{-1}(T)$, but one has to be aware that these common notations may be ambiguous in certain situations.

References

N. Bourbaki, "Elements of mathematics. Theory of sets" , Addison-Wesley (1968) (Translated from French)

Paul R. Halmos, Naive Set Theory.
   (The University Series in Undergraduate Mathematics) Princeton, N. J., etc., Van Nostrand, 1960.
   Reprinted: (Undergraduate Texts in Mathematics) New York, etc., Springer, 1974.

How to Cite This Entry:
Mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mapping&oldid=15615
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article