Set notation: Difference between revisions

Revision as of 05:42, 29 November 2013

Revision as of 01:53, 9 January 2014

Line 19:

The simplest notational approach of this type, which is feasible only for fairly small sets, is to enumerate the elements exhaustively. Thus the set of suits in a standard deck of playing cards is denoted by {♠,&nbsp;<span style="color:red">♦</span>,&nbsp;<span style="color:red">♥</span>,&nbsp;♣} and the set of even [[prime numbers]] is denoted by {{math|&#123;2&#125;}}. This approach also provides the notation {{math|&#123;&#125;}} for the empty set.

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|&#123;''π'', 6, 1/2&#125;}} and {{math|&#123;1/2, ''π'', 6&#125;}} are two enumerations of a single set. Likewise, repeated mention of an element is also irrelevant, so {{math|&#123;1, 2, 2, 3, 3, 3&#125; {{=}} &#123;1, 2, 3&#125;}} To deal with collections for which members' multiplicity ''is'' significant, there is a generalization of sets called ''[[multisetsmultiset]]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|&#123;1, 2, 3, ..., 10&#125;}}. Here, of course, the ellipsis means "and so forth." Note that wherever 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|&#123;1, ..., ''n''²&#125;}}.

Some [[infinite setsset]]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|&#123;1, 2, 3, ...&#125;}}. 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|&#123;0, 1, 3, ..., ''k''(''k''-1)/2, ...&#125;}}.

A more powerful mechanism for denoting a set in terms of its elements is [[set-builder notation]]. Here the general pattern is {{math|&#123;''x'' : ''P''(''x'')&#125;}}, 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|&#123;(''u'', ''v'') : (''u''−''a'')² + (''v''-''b'')² {{=}} ''r''²&#125;}}.