Set notation: Difference between revisions

Revision as of 20:13, 19 November 2017

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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>.

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''.

There are many other examples of this convention. One is the set of [[Function (mathematics)|functions]] from set {{math|''A''}} to set {{math|''B''}}. When {{math|''A''}} and {{math|''B''}} are [[Finite set|finite]], specifying any such function amounts to choosing for each element of {{math|''A''}} which element of {{math|''B''}} should be its [[image (mathematics)|image]]. So, the number of these functions is {{math|&#124;''B''&#124;^{&#124;''A''&#124;}}}. Thus, one denotes the set of all functions from {{math|''A''}} to {{math|''B''}} as {{math|''B''^''A''}}. Another example is the [[power set]] of a set {{math|''S''}}, which, having cardinality {{math|2^{&#124;''S''&#124;}}}, is denoted by {{math|2^''S''}}. Note, though, that since any [[subset]] of {{math|''S''}} may be seen as a function assigning to each element of {{math|''S''}} one or the other element of {include,&nbsp;exclude}, the notation {{math|2^''S''}} may be seen as a special case of {{math|''B''^''A''}}. The cardinality metaphor has also been used to derive from the standard notation for [[binomial coefficients]] the notation <math>\tbinom X k</math> for the set of all {{math|''k''}}-element subsets of a set {{math|''X''}}.