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Another difficulty with the idea of a universal set concerns the power set of the set of all sets. Because this power set is a set of sets, it would necessarily be a subset of the set of all sets, provided that both exist. However, this conflicts with Cantor's theorem that the power set of any set (whether infinite or not) always has strictly ...
In mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. [1] In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. [2]
In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set, the set of all subsets of , known as the power set of , has a strictly greater cardinality than itself.
The simple theorems in the algebra of sets are some of the elementary properties of the algebra of union (infix operator: ∪), intersection (infix operator: ∩), and set complement (postfix ') of sets. These properties assume the existence of at least two sets: a given universal set, denoted U, and the empty set, denoted {}.
The empty set is also occasionally called the null set, [11] though this name is ambiguous and can lead to several interpretations. The power set of a set A, denoted (), is the set whose members are all of the possible subsets of A. For example, the power set of {1, 2} is { {}, {1}, {2}, {1, 2} }.
8 Ways of defining sets/Relation to descriptive set theory. 9 More general objects still called sets. ... Countable set, Uncountable set; Power set; Relative to a ...
For instance, when investigating properties of the real numbers R (and subsets of R), R may be taken as the universal set. A true universal set is not included in standard set theory (see Paradoxes below), but is included in some non-standard set theories. Given a universal set U and a subset A of U, the complement of A (in U) is defined as
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.