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The power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers (see Cardinality of the continuum). The power set of a set S, together with the operations of union, intersection and complement, is a Σ-algebra over S and can be viewed as the prototypical example of a Boolean algebra.
The power set axiom does not specify what subsets of a set exist, only that there is a set containing all those that do. [2] Not all conceivable subsets are guaranteed to exist. In particular, the power set of an infinite set would contain only "constructible sets" if the universe is the constructible universe but in other models of ZF set ...
The following proposition says that for any set , the power set of , ordered by inclusion, is a bounded lattice, and hence together with the distributive and complement laws above, show that it is a Boolean algebra.
The set of all subsets of a given set is called the power set of and is denoted by ℘ (). The power set ℘ of a given set is a family of sets over .. A subset of having elements is called a -subset of .
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.
If is the whole power set of then () is called a full complex algebra or power algebra. Every (normal) Boolean algebra with operators can be represented as a field of sets on a relational structure in the sense that it is isomorphic to the complex algebra corresponding to the field.
A finite set with elements has distinct subsets. That is, the power set ℘ of a finite set S is finite, with cardinality | |. Any subset of a finite set is finite. The set of values of a function when applied to elements of a finite set is finite.
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 ...