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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'')<sup>2</sup> + (''v''-''b'')<sup>2</sup> {{=}} ''r''<sup>2</sup>}}}.
A notable exception to the braces notation is used to express [[interval (mathematics)|intervals]] on the [[real line]]. Any such interval is well defined only because the real numbers are [[Total order|totally ordered]]. It 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 ==
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