Set notation: Difference between revisions

Revision as of 00:02, 2 August 2019

Revision as of 12:49, 2 August 2019

Line 23:

The [[semantics]] of the term ''set'' imposes certain [[syntactic]] constraints on these notations. The only information that is fundamental for a set is which particular objects are, or are not, elements. As a result, the order in which elements appear in an enumeration is irrelevant: {{math|{''π'', 6, 1/2}}} and {{math|{1/2, ''π'', 6}}} are two enumerations of the same set. Likewise, repeated mention of an element is also irrelevant, so {{math|{1, 2, 2, 3, 3, 3} {{=}} {1, 2, 3}}}. To deal with collections for which members' multiplicity ''is'' significant, there is a generalization of sets called ''[[multiset]]s''.

A variant of this explicitly exhaustive enumeration uses ranges of elements and features the [[ellipsis]]. If <math>m</math> and <math>n</math> are natural numbers, <math>\{m, \dots, n\}</math> denotes the set <math>\{i\in\mathbb N\mid m\leq i\leq n\}</math>. For example, the set of the first ten natural numbers can be written as {{math|&#123;1, ..., 10&#125;}}. If <math>m>n</math> then <math>\{m, \dots, n\}</math> is the empty set <math>\emptyset</math>. Here, of course, the ellipsis means "and so forth." Wherever an ellipsis is used to denote a range, it is punctuated as though it were an element of the set. If either extreme of a range is indeterminate, it may be denoted by a [[mathematical expression]] giving a formula to compute it. As an example, if {{math|''n''}} is known from context to be a positive integer, then the set of the first {{math|''n''}} [[square number|perfect squares]] may be denoted by {{math|&#123;1, 4, ..., ''n''²&#125;}}.

In general, if <math>n</math> is a natural number, then <math>\{1, \dots, n\}</math> denotes the set <math>\{i\in\mathbb N\mid 1\leq i\leq n\}</math>. A subtle special is <math>n=0</math>, in which <math>\{1, \dots, 0\}</math> is the empty set <math>\emptyset</math>.

Some [[infinite set]]s, too, can be represented in this way. An example is denoting the set of natural numbers (for which one notation described above is {{math|'''N'''}}) by {{math|&#123;1, 2, 3, ...&#125;}}. In cases where the infinitely repeating pattern is not obvious, one can insert an expression to represent a generic element of the set, as with {{math|&#123;0, 1, 3, ..., ''k''(''k''-1)/2, ...&#125;}}.