Editing Set-builder notation (section)

=== Examples ===
The following examples illustrate particular sets defined by set-builder notation via predicates. In each case, the domain is specified on the left side of the vertical bar, while the rule is specified on the right side. 
* <math>\{ x \in \mathbb{R}   \mid  x > 0\}</math> is the set of all strictly [[positive number|positive]] [[real number]]s, which can be written in [[interval notation]] as <math>(0, \infty)</math>.
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* <math>\{ x \in \mathbb{R}  \mid |x| = 1 \}</math> is the set <math>\{-1, 1\}</math>. This set can also be defined as <math>\{x \in \mathbb{R} \mid x^2 = 1\}</math>; see [[#Equivalent predicates yield equal sets|equivalent predicates yield equal sets]] below.
* For each integer {{math|''m''}}, we can define <math>G_m = \{x  \in \mathbb{Z}  \mid  x \ge m \} = \{ m, m + 1, m + 2, \ldots\}</math>. As an example, <math>G_3 = \{x \in \mathbb{Z} \mid x \ge 3 \} = \{ 3, 4, 5, \ldots\}</math> and <math>G_{-2} = \{ -2, -1, 0, \ldots\}</math>.
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* <math>\{ (x,y) \in \mathbb{R} \times \mathbb{R} \mid 0 < y < f(x) \}</math>  is the set of pairs of real numbers such that ''y'' is greater than 0 and less than {{math|''f''(''x'')}}, for a given [[function (mathematics)|function]] {{math|''f''}}.   Here the [[cartesian product]] <math>\mathbb{R}\times\mathbb{R}</math> denotes the set of ordered pairs of real numbers. 
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* <math>\{n   \in \mathbb{N} \mid  (\exists k)  [k\in \mathbb{N}\land n = 2k]  \} </math> is the set of all [[even number|even]] [[natural number]]s. The <math>\land</math> sign stands for "and", which is known as [[logical conjunction]]. The &exist; sign stands for "there exists", which is known as [[existential quantification]]. So for example, <math> (\exists x) P(x)</math> is read as  "there exists an {{math|''x''}} such that {{math|''P''(''x'')}}".
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*  <math>\{n \mid  (\exists k \in \mathbb{N} )  [n = 2k]  \} </math> is a notational variant for the same set of even natural numbers. It is not necessary to specify that {{math|''n''}} is a natural number, as this is implied by the formula on the right.
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* <math>\{a \in \mathbb{R} \mid (\exists p\in \mathbb{Z} )(\exists q\in \mathbb{Z} )[ q \not = 0 \land aq=p] \}</math> is the set of [[rational number]]s; that is, real numbers that can be written as the ratio of two [[integer]]s.