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== Metaphor in denoting sets ==
Since so much of mathematics consists in discovering and exploiting patterns, it is perhaps not surprising that there should have arisen various set-denotational conventions that strike practitioners as obvious or natural—if sometimes only once the pattern has been pointed out.
One class comprises those notations deriving the symbol for a set from the algebraic form of a representative element of the set. As an example, consider the set of even numbers. Since a number {{math|''b''}} is even precisely if there exists some integer {{math|''a''}} such that {{math|''b'' {{=}} 2''a''}}, the following [[Set-builder_notation#Terms_more_complicated_than_a_single_variable|variation on set-builder notation]] could be used to denote this set: {{math|{2''a'' : ''a''∈'''Z'''}}} (compare this with the formal set-builder notation: {{math|{''b''∈'''Z''' : ∃ ''a''∈'''Z''': ''b'' {{=}} 2''a''}}}). Alternatively, a single symbol for the set of even numbers is {{math|2'''Z'''}}. Likewise, since any odd number must have the form {{math|2''a'' + 1}} for some integer {{math|''a''}}, the set of odd numbers may be denoted {{math|2'''Z'''+1}}.
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