翻訳と辞書 Words near each other ・ Alger, Washington ・ Alger-Sullivan Lumber Company Residential Historic District ・ Algeria ・ Algeria A' national football team ・ Algebra ・ Algebra & Number Theory ・ Algebra (disambiguation) ・ Algebra (singer) ・ Algebra bundle ・ Algebra Colloquium ・ Algebra homomorphism ・ Algebra i Logika ・ Algebra of Communicating Processes ・ Algebra of physical space ・ Algebra of random variables ・ Algebra of sets ・ Algebra over a field ・ Algebra over an operad ・ Algebra Project ・ Algebra representation ・ Algebra tile ・ Algebra Universalis ・ Algebraic ・ Algebraic & Geometric Topology ・ Algebraic analysis ・ Algebraic character ・ Algebraic closure ・ Algebraic cobordism ・ Algebraic code-excited linear prediction ・ Algebraic combinatorics Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Algebra of sets ： ウィキペディア英語版 Algebra of sets The algebra of sets defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations. Any set of sets closed under the set-theoretic operations forms a Boolean algebra with the join operator being ''union'', the meet operator being ''intersection'', and the complement operator being ''set complement''. ==Fundamentals== The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset". It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory, and for a full rigorous axiomatic treatment see axiomatic set theory. ==The fundamental laws of set algebra== The binary operations of set union ($\backslash cup$) and intersection ($\backslash cap$) satisfy many identities. Several of these identities or "laws" have well established names. :Commutative laws: :: *$A\; \backslash cup\; B\; =\; B\; \backslash cup\; A\backslash ,\backslash !$ :: *$A\; \backslash cap\; B\; =\; B\; \backslash cap\; A\backslash ,\backslash !$ :Associative laws: :: *$(A\; \backslash cup\; B)\; \backslash cup\; C\; =\; A\; \backslash cup\; (B\; \backslash cup\; C)\backslash ,\backslash !$ :: *$(A\; \backslash cap\; B)\; \backslash cap\; C\; =\; A\; \backslash cap\; (B\; \backslash cap\; C)\backslash ,\backslash !$ :Distributive laws: :: *$A\; \backslash cup\; (B\; \backslash cap\; C)\; =\; (A\; \backslash cup\; B)\; \backslash cap\; (A\; \backslash cup\; C)\backslash ,\backslash !$ :: *$A\; \backslash cap\; (B\; \backslash cup\; C)\; =\; (A\; \backslash cap\; B)\; \backslash cup\; (A\; \backslash cap\; C)\backslash ,\backslash !$ The analogy between unions and intersections of sets, and addition and multiplication of numbers, is quite striking. Like addition and multiplication, the operations of union and intersection are commutative and associative, and intersection ''distributes'' over unions. However, unlike addition and multiplication, union also distributes over intersection. Two additional pairs of laws involve the special sets called the empty set Ø and the universal set $U$; together with the complement operator (''A''C denotes the complement of ''A''). The empty set has no members, and the universal set has all possible members (in a particular context). :Identity laws: :: *$A\; \backslash cup\; \backslash varnothing\; =\; A\backslash ,\backslash !$ :: *$A\; \backslash cap\; U\; =\; A\backslash ,\backslash !$ :Complement laws: :: *$A\; \backslash cup\; A^C\; =\; U\backslash ,\backslash !$ :: *$A\; \backslash cap\; A^C\; =\; \backslash varnothing\backslash ,\backslash !$ The identity laws (together with the commutative laws) say that, just like 0 and 1 for addition and multiplication, Ø and U are the identity elements for union and intersection, respectively. Unlike addition and multiplication, union and intersection do not have inverse elements. However the complement laws give the fundamental properties of the somewhat inverse-like unary operation of set complementation. The preceding five pairs of laws—the commutative, associative, distributive, identity and complement laws—encompass all of set algebra, in the sense that every valid proposition in the algebra of sets can be derived from them. Note that if the complement laws are weakened to the rule $(A^C)^C\; =\; A$, then this is exactly the algebra of propositional linear logic. 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「Algebra of sets」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.