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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 the simplest version of the powerset construction, the set of all states of the DFA is the powerset of Q, the set of all possible subsets of Q. However, many states of the resulting DFA may be useless as they may be unreachable from the initial state. An alternative version of the construction creates only the states that are actually reachable.
Counting the empty set as a subset, a set with elements has a total of subsets, and the theorem holds because > for all non-negative integers. Much more significant is Cantor's discovery of an argument that is applicable to any set, and shows that the theorem holds for infinite sets also.
A generalized form of the diagonal argument was used by Cantor to prove Cantor's theorem: for every set S, the power set of S—that is, the set of all subsets of S (here written as P(S))—cannot be in bijection with S itself. This proof proceeds as follows: Let f be any function from S to P(S). It suffices to prove f cannot be surjective.
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 power set of is the set of all subsets of and will be denoted ... the power set of , ordered by inclusion, is a bounded lattice, and hence ...
Abbreviation for "power (set)" power "Power" is an archaic term for cardinality power set powerset The powerset or power set of a set is the set of all its subsets pre-ordering A relation that is reflexive and transitive but not necessarily antisymmetric, allowing for the comparison of elements in a set. primitive recursive set
The set of all subsets of is called its power set, and is denoted by (). [6] The inclusion relation is a partial order on the set ... A finite set A is a subset of B, ...