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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 a single 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]]. Thus the set of the first ten natural numbers is {{math|{1, 2, 3, ..., 10}}}. Here, of course, the ellipsis means "and so forth." Note that 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|{1, 4, ..., ''n''<sup>2</sup>}}}.
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|{1, 2, 3, ...}}}. 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|{0, 1, 3, ..., ''k''(''k''-1)/2, ...}}}.
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