Set notation: Difference between revisions

Revision as of 11:25, 15 June 2017

Revision as of 11:36, 15 June 2017

Line 27:

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]]. AnyIt suchmakes intervaluse is well defined only becauseof the realfact numbersthat areany [[Totalsuch order|totally ordered]]. Itinterval 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 this 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 ==