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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 ...
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".
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 ...
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.
1. The difference of two sets: x~y is the set of elements of x not in y. 2. An equivalence relation \ The difference of two sets: x\y is the set of elements of x not in y. − The difference of two sets: x−y is the set of elements of x not in y. ≈ Has the same cardinality as × A product of sets / A quotient of a set by an equivalence ...
Given any set , an equivalence relation over the set [] of all functions can be obtained as follows. Two functions are deemed equivalent when their respective sets of fixpoints have the same cardinality, corresponding to cycles of length one in a permutation.
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 a set is completely determined by its elements; the description is immaterial. For example, the set with elements 2, 3, and 5 is equal to the set of all prime numbers less than 6. If the sets A and B are equal, this is denoted symbolically as A = B (as usual).