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Definition of Sets

A set must be well defined; i.e., for any given object, it must be unambiguous whether or not the object is an element of the set. For example, if a set contains all the chairs in a designated room, then any chair can be determined either to be in or not in the set. If there were no chairs in the room, the set would be called the empty, or null, set, i.e., one containing no elements. A set is usually designated by a capital letter. If A is the set of even numbers between 1 and 9, then A = {2, 4, 6, 8}. The braces, {}, are commonly used to enclose the listed elements of a set. The elements of a set may be described without actually being listed. If B is the set of real numbers that are solutions of the equation x 2 = 9, then the set can be written as B = { x : x 2 = 9} or B = { x | x 2 = 9}, both of which are read: B is the set of all x such that x 2 = 9; hence B is the set {3, - 3}.

Membership in a set is indicated by the symbol ∈ and nonmembership by ∉; thus, xA means that element x is a member of the set A (read simply as " x is a member of A ") and yA means y is not a member of A. The symbols ⊂ and ⊃ are used to indicate that one set A is contained within or contains another set B; AB means that A is contained within, or is a subset of, B; and AB means that A contains, or is a superset of, B.

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The Columbia Electronic Encyclopedia, 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.

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