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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 () and intersection () satisfy many identities. Several of these identities or "laws" have well established names. :Commutative laws: :: * :: * :Associative laws: :: * :: * :Distributive laws: :: * :: * 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 ; 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: :: * :: * :Complement laws: :: * :: * 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 , 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
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