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| + | {{TEX|done}} |
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| $ | | $ |
| \def\P{\mathcal P} % power set | | \def\P{\mathcal P} % power set |
− | \def\iff{\Longleftrightarrow} | + | \def\iff{\Leftrightarrow} |
| $ | | $ |
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| The mapping is usually denoted as $ f : A \to B $, and $ a \mapsto f(a) $ | | 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 $a$. | | 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 $. |
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| ''Remark:''<br> | | ''Remark:''<br> |
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| For $ f=(A,B,G_f) $ and $ g=(C,D,G_g) $ with $ B \subset C $ | | 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,C,G) $ with | + | 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 \} $. | + | : $ 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 ) \} $. |
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| ''Remarks:'' <br> | | ''Remarks:'' <br> |
− | (a) The condition $ B \subset C $ can be relaxed to $ f(B) \subset C $. | + | (a) The condition $ B \subset C $ can be relaxed to $ f(A) \subset C $. |
| <br> | | <br> |
| (b) If only graphs are used then the graph of the composition is defined (as above) by | | (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 \} $ | + | : $ 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. | | but may turn out to be empty. |
| | | |
| ==== Induced mappings ==== | | ==== Induced mappings ==== |
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− | Every mapping $ f : A \to B $ induces two mappings between the power sets $\P(A)$ and $\P(B)$. | + | Every mapping $ f : A \to B $ induces two mappings between the [[power set]]s $\P(A)$ 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 $ | | : $ f_\ast : \P(A) \to \P(B) $ defined by $ f_\ast (S) := \{ f(a) \mid a \in S \}$ for $ S \subset A $ |
| and | | and |
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| $ f^\ast (T) $ is called the ''inverse image'' of $T$ under $f$, usually denoted as $f^{-1}(T)$, | | $ 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. | | but one has to be aware that these common notations may be ambiguous in certain situations. |
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− | ''single-valued''
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− | 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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− | 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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− | 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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− | 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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− | <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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− | 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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− | 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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− | 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]]).
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− | 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.
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| ====References==== | | ====References==== |
− | <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>
| + | N. Bourbaki, "Elements of mathematics. Theory of sets" , Addison-Wesley (1968) (Translated from French) |
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− | | + | Paul R. Halmos, ''Naive Set Theory.'' |
− | | + | <br> (The University Series in Undergraduate Mathematics) Princeton, N. J., etc., Van Nostrand, 1960. |
− | ====Comments====
| + | <br> ''Reprinted'': (Undergraduate Texts in Mathematics) New York, etc., Springer, 1974. |
− | 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]]].
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− | ====References====
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− | <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> | |