Mappings

Janovec, V.; Hahn, Th.; Klapper, H.

doi:10.1107/97809553602060000643

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 3.2, pp. 380-381

Section 3.2.3.1.3. Mappings

V. Janovec,^a ^* Th. Hahn^b and H. Klapper^c

^a Department of Physics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic,^bInstitut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D-52056 Aachen, Germany, and ^cMineralogisch-Petrologisches Institut, Universität Bonn, D-53113 Bonn, Germany
Correspondence e-mail: janovec@fzu.cz

3.2.3.1.3. Mappings

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A mapping $[\varphi]$ of a set $[{\sf A}]$ into a set $[{\sf B}]$ is a rule which assigns to each element $[{\bf S} \in {\sf A}]$ a unique element $[{\bf M} \in {\sf B}]$ . This is written symbolically as $[\varphi:{\bf S} \,\mapsto\,{ \bf M}]$ or $[{\bf M}= \varphi({\bf S})]$ , and one says that $[\bf{S}]$ is mapped to $[\bf{M}]$ under the mapping $[\varphi]$ . The element $[\bf{M}]$ is called the image of the element S under $[\varphi]$ . The assignment $[\varphi:{\bf S} \,\mapsto\, {\bf M}]$ can be expressed by an ordered pair $[({\bf S},{\bf M})]$ , if one ascribes $[\bf{S}]$ to the first member of the pair and the element $[\bf{M}]$ to the second member of the pair $[({\bf S},{\bf M})]$ . Then the mapping $[\varphi]$ of a set $[{\sf A}]$ into a set $[{\sf B}]$ , symbolically written as $[\varphi:{\sf A} \rightarrow {\sf B}]$ , can be identified with such a subset of ordered pairs of the Cartesian product $[{\sf A} \times {\sf B}]$ in which each element $[\bf{S}]$ of $[{\sf A}]$ occurs exactly once as the first member of the pair $[({\bf S},{\bf M})]$ . If $[{\sf A}]$ is a finite set, then $[\varphi]$ consists of $[|{\sf A}|]$ ordered pairs.

We note that in a mapping $[\varphi:{\sf A} \rightarrow {\sf B}]$ several elements of $[{\sf A}]$ may be mapped to the same element of $[{\sf B}]$ . In such a case, the mapping $[\varphi]$ is called a many-to-one mapping. If the mapping $[\varphi:{\sf A} \rightarrow {\sf B}]$ is such that each element of $[{\sf B}]$ is the image of some element of $[{\sf A}]$ , then the mapping $[\varphi]$ is called a mapping of $[{\sf A}]$ onto $[{\sf B}]$ . If $[\varphi]$ is a mapping of $[{\sf A}]$ onto $[{\sf B}]$ and, moreover, each element of $[{\sf B}]$ is the image of exactly one element of $[{\sf A}]$ , then the mapping $[\varphi]$ becomes a one-to-one correspondence between $[{\sf A}]$ and $[{\sf B}]$ , $[\varphi:{\sf A} \leftrightarrow {\sf B}]$ . In this case, $[{\sf A}]$ and $[{\sf B}]$ are of the same order.

One often encounters a situation in which one assigns to each ordered pair $[({\bf S},{\bf M})]$ an element $[\bf{N}]$ , where all three elements $[{\bf S}, {\bf M}, {\bf N}]$ are elements from the same set $[{\sf A}]$ , symbolically $[\varphi:({\bf S},{\bf M}) \,\mapsto\, {\bf N}]$ ; $[{\bf S}, {\bf M}, {\bf N} \in {\sf A}]$ or $[\varphi:{\sf A} \times {\sf A} \rightarrow {\sf A}]$ . Such a mapping is called a binary operation or a composition law on the set $[{\sf A}]$ . A sum of two numbers [a+b=c] or a product of two numbers $[a\cdot b=c]$ , where [a,b,c] belong to the set of all real numbers, are elementary examples of binary operations.

References

International Tables for Crystallography (2006). Vol. D. ch. 3.2, pp. 380-381