Set notation: Difference between revisions

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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 comprisesconsists of 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}}.

A second class is based on a strong logical relationship between a set and a particular integer. One example is the bracket notation, in which the set {{math|&#123;1, ..., ''n''&#125;}} of the first {{math|''n''}} positive integers is denoted by {{math|[''n'']}}. (As a related point, when endowed with the standard less-than-or-equal [[Relation (mathematics)|relation]] {{math|≤}}, the set {{math|[''n'']}} yields the [[poset]] denoted by {{math|'''''n'''''}}.) Another example arises from [[modular arithmetic]], where [[equivalence classes]] are denoted by <math>\bar{a}_n</math>, which may be understood to represent the set of integers that leave remainder {{math|''a''}} on division by {{math|''n''}}. Thus yet another notation for the set of even numbers is <math>\bar{0}_2</math>.