Set notation: Difference between revisions

Revision as of 21:34, 8 December 2018

Revision as of 16:02, 9 December 2018

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The [[semantics]] of the term ''set'' imposes certain [[syntactic]] constraints on these notations. The only information that is fundamental for a set is which particular objects are, or are not, elements. As a result, the order in which elements appear in an enumeration is irrelevant: {{math|{''π'', 6, 1/2}}} and {{math|{1/2, ''π'', 6}}} are two enumerations of the same set. Likewise, repeated mention of an element is also irrelevant, so {{math|{1, 2, 2, 3, 3, 3} {{=}} {1, 2, 3}}}. To deal with collections for which members' multiplicity ''is'' significant, there is a generalization of sets called ''[[multiset]]s''.

A variant of this explicitly exhaustive enumeration uses ranges of elements and features the [[ellipsis]]. Thus the set of the first ten natural numbers is {{math|{1, 2, 3, ..., 10}}}. Here, of course, the ellipsis means "and so forth." Note that whereverWherever an ellipsis is used to denote a range, it is punctuated as though it were an element of the set. If either extreme of a range is indeterminate, it may be denoted by a [[mathematical expression]] giving a formula to compute it. As an example, if {{math|''n''}} is known from context to be a positive integer, then the set of the first {{math|''n''}} [[square number|perfect squares]] may be denoted by {{math|{1, 4, ..., ''n''²}}}.

Some [[infinite set]]s, too, can be represented in this way. An example is denoting the set of natural numbers (for which one notation described above is {{math|'''N'''}}) by {{math|{1, 2, 3, ...}}}. In cases where the infinitely repeating pattern is not obvious, one can insert an expression to represent a generic element of the set, as with {{math|{0, 1, 3, ..., ''k''(''k''-1)/2, ...}}}.

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A more powerful mechanism for denoting a set in terms of its elements is [[set-builder notation]]. Here the general pattern is {{math|{''x'' : ''P''(''x'')}}}, which denotes the set of all elements {{math|''x''}} (from some [[universal set]]) for which the assertion {{math|''P''(''x'')}} about {{math|''x''}} is true. For example, when understood as a set of points, the circle with radius {{math|''r''}} and center {{math|(''a'', ''b'')}}, may be denoted as {{math|{(''u'', ''v'') : (''u''−''a'')² + (''v''-''b'')² {{=}} ''r''²}}}.

A notable exception to the braces notation is used to express [[interval (mathematics)|intervals]] on the [[real line]]. It makes use of the fact that any such interval is completely determined by its left and right endpoints: the [[unit interval]], for instance, is the set of reals between 0 and 1 (inclusive). The convention for denoting intervals uses brackets and parentheses, depending as the corresponding endpoint is included in or excluded from the set, respectively. Thus the set of reals with [[absolute value]] less than one is denoted by {{math|(−1, 1)}} — note that thisThis is very different from the [[ordered pair]] with first entry −1 and second entry 1. As other examples, the set of reals {{math|''x''}} that satisfy {{math|2 < ''x'' ≤ 5}} is denoted by {{math|(2, 5]}}, and the set of nonnegative reals is denoted by {{math|[0, ∞)}}.

== Metaphor in denoting sets ==