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Assuming the existence of an infinite set N consisting of all natural numbers and assuming the existence of the power set of any given set allows the definition of a sequence N, P(N), P(P(N)), P(P(P(N))), … of infinite sets where each set is the power set of the set preceding it. By Cantor's theorem, the cardinality of each set in this ...
Two sets [twelve-tone series], P and P ′ will be considered equivalent [equal] if and only if, for any p i,j of the first set and p ′ i ′,j ′ of the second set, for all is and js [order numbers and pitch class numbers], if i=i ′, then j=j ′. (= denotes numeral equality in the ordinary sense).
The set of the equivalence classes is sometimes called the quotient set or the quotient space of by , and is denoted by /. When the set S {\displaystyle S} has some structure (such as a group operation or a topology ) and the equivalence relation ∼ {\displaystyle \,\sim \,} is compatible with this structure, the quotient set often inherits a ...
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".
This article lists mathematical properties and laws of sets, involving 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.
In set theory, any two sets are defined to be equal if they have all the same members. This is called the Axiom of extensionality. Usually set theory is defined within logic, and therefore uses the equality described above, however, if a logic system does not have equality, it is possible to define equality within set theory.
In set theory, the axiom of extensionality states that two sets are equal if and only if they contain the same elements. In mathematics formalized in set theory, it is common to identify relations—and, most importantly, functions —with their extension as stated above, so that it is impossible for two relations or functions with the same ...
Thus it is meaningful to speak of a "presentation of an equivalence relation," i.e., a presentation of the corresponding groupoid; Bundles of groups, group actions, sets, and equivalence relations can be regarded as special cases of the notion of groupoid, a point of view that suggests a number of analogies;