Set notation: Difference between revisions

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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}}.

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'']}}. (asAs 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>.

Another set-denotational convention that relies on metaphor comes from [[enumerative combinatorics]]. It derives a symbol for a set {{math|''S''}} from an expression for the set's [[cardinality]], or size, {{math|&#124;''S''&#124;}}. Perhaps the simplest and best known example is the [[Cartesian product]] of sets {{math|''A''}} and {{math|''B''}}, which is the set {{math|&#123;(''a'', ''b'') : ''a''∈''A'', ''b''∈''B''&#125;}}. Since, in this set, every element of {{math|''A''}} gets paired exactly once with every element of {{math|''B''}}, its cardinality is {{math|&#124;''A''&#124; × &#124;''B''&#124;}}. For this reason, the set is denoted by {{math|''A''×''B''}}. In fact, that same fact about its cardinality is why this set is called a ''product''.