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In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by stating the properties that its members must satisfy. Forming sets in this manner is also known as set comprehension, set abstraction or as defining a set's intension. Although some simply refer to it as ''set notation,'' that label may be better reserved for the broader class of means of denoting sets. == Direct, ellipses, and informally specified sets == A set is an unordered list of ''elements''. (An ''element'' may also be referred to as a ''member''). An element may be any mathematical entity. We can denote a set directly by listing all of its elements between curly brackets, as in the following two examples: * is a set holding the four numbers 3, 7, 15, and 31. * is the set containing 'a','b', and 'c'. When it is desired to denote a set that contains elements from a regular sequence an ellipses notation may be employed, as shown in the next two examples: * is the set of integers between 1 and 100 inclusive. * is the set of natural numbers.〔 use an ellipsis to informally define the natural numbers: 'Intuitively, the set ℕ = of all ''natural numbers'' may be described as follows: ℕ contains an "initial" number 0; ...'. They follow that with their version of the Peano Postulates. (p. 15)〕 There is no order among the elements of a set, but with the ellipses notation we show an ordered sequence before the ellipsis as a convenient notational vehicle for explaining to a reader which members are in a set. The first few elements of the sequence are shown then the ellipses indicate that the simplest interpretation should be applied for continuing the sequence. Should no terminating value appear to the right of the ellipses then the sequence is considered to be unbounded. Mathematicians sometimes denote a set using general prose, as shown in the following example. * is the set of all addresses on Pine Street. The meaning of this prose must be clear to the reader or the mathematician who wrote it has failed to sufficiently define the set for the reader. The ellipses and simple prose approaches give the reader rules for building the set rather than directly presenting the elements. Mathematicians find this approach of providing building rules to be convenient and important so they have extended and formalized the set builder notation as further described in this article. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Set-builder notation」の詳細全文を読む スポンサード リンク
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