Sets

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, p. 380

Section 3.2.3.1.1. Sets

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

3.2.3.1.1. Sets

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Definition 3.2.3.1. A set is a collection of distinguishable objects. The objects constituting a set are called elements (or points) of the set.

In Chapter 3.4 we encounter mainly two types of sets: sets the elements of which are crystalline objects (domain states, domain twins, domain walls etc.), and sets, like groups, with elements of mathematical nature, e.g. rotations, transformations, operations etc. The sets of crystalline objects will be denoted by capital sans-serif letters, e.g. $[{\sf A},{\sf B},\ldots]$ , and capital bold letters, e.g. $[\bf{S}, \bf{M}, \bf{N},\ldots]$ or $[{\bf S}_1, {\bf S}_2, {\bf S}_3,\ldots]$ , will be used to denote elements of such sets. Groups will be denoted by capital italic letters, e.g. G, F etc., and their elements by lower-case italic letters, e.g. $[g, h,\ldots]$ . The exposition of this section is given for sets the elements of which are (crystalline) objects, but all notions and relations hold for any other sets.

If an element $[\bf{S}]$ belongs to the set $[{\sf A}]$ , one writes $[{\bf S} \in {\sf A}]$ , in the opposite case $[{\bf S} \not\in {\sf A}]$ . Sets consisting of a small number of elements can be expressed explicitly by writing their elements between curly braces, $[{\sf A}=\{{\bf S}, {\bf M}, {\bf N}, {\bf Q}\}]$ . The order of elements in the symbol of the set is irrelevant. From the definition of a set it follows that there are no equal elements in the set, or in other words, any two equal elements coalesce into one: $[\{{\bf S},{\bf S}\}=\{{\bf S}\}.\eqno(3.2.3.1)]$ If a set contains many (or an infinite number of) elements, the elements are specified in another way, e.g. by stating that they have a certain property in common.

The number of elements in a set is the order of the set. A finite set $[{\sf A}]$ consists of a finite number of elements and this number is denoted by $[|{\sf A}|]$ . An infinite set contains infinite number of elements and an empty set, denoted by $[\emptyset]$ , contains no element. In what follows, the term `set' will mean a `finite nonempty set' unless explicitly stated otherwise.

A set $[{\sf B}]$ is a subset of $[{\sf A}]$ , $[{\sf B} \subseteq {\sf A}]$ or $[{\sf A} \supseteq {\sf B} ]$ , if every element of $[{\sf B}]$ is an element of $[{\sf A}]$ . If each element of $[{\sf B}]$ is an element of $[{\sf A}]$ , and vice versa, then $[{\sf B}]$ is equal to or identical with $[{\sf A}]$ , $[{\sf B}={\sf A}]$ or $[{\sf A}={\sf B}]$ . If there exists at least one element of $[{\sf A}]$ which is not contained in $[{\sf B}]$ , then $[{\sf B}]$ is a proper subset of $[{\sf A}]$ , $[{\sf B} \subset {\sf A}]$ or $[{\sf A} \supset {\sf B}]$ . The subset $[{\sf B}]$ is often defined by a restriction that specifies only some elements of $[{\sf A}]$ as elements of $[{\sf B}]$ . This is written in short as $[{\sf B} =]$ $[\{{\bf S} \in {\sf A}|\hbox{restriction on } {\bf S}\}]$ ; the expression means that $[{\sf B}]$ consists of all elements of $[{\sf A}]$ that satisfy the restriction given behind the sign |.

The intersection of two sets $[{\sf A}]$ and $[{\sf B}]$ , $[{\sf A} \cap {\sf B}]$ or $[{\sf B} \cap {\sf A}]$ , is a set comprising all elements that belong both to $[{\sf A}]$ and to $[{\sf B}]$ . If the sets $[{\sf A}]$ and $[{\sf B}]$ have no element in common, $[{\sf A} \cap {\sf B} = \emptyset]$ , then one says that the sets $[{\sf A}]$ and $[{\sf B}]$ are disjoint. The union of sets $[{\sf A}]$ and $[{\sf B}]$ , $[{\sf A} \cup {\sf B}]$ or $[{\sf B} \cup {\sf A}]$ , is a set consisting of all elements that belong either to $[{\sf A}]$ or to $[{\sf B}]$ . Sometimes the symbol [+] is used instead of the symbol $[\cup]$ . The difference of set $[{\sf A}]$ and $[{\sf B}]$ , or the complement of $[{\sf B}]$ in $[{\sf A}]$ , $[{\sf A}-{\sf B}]$ , comprises those elements of $[{\sf A}]$ that do not belong to $[{\sf B}]$ .

References

International Tables for Crystallography (2006). Vol. D. ch. 3.2, p. 380