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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]
A is a subset of B (denoted ) and, conversely, B is a superset of A (denoted ). In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B.
The algebra of sets describes the properties of all possible subsets of U, called the power set of U and denoted P(U). P ( U ) is assumed closed under union, intersection, and set complement. The algebra of sets is an interpretation or model of Boolean algebra , with union, intersection, set complement, U , and {} interpreting Boolean sum ...
A more useful example is the set of subsets of the real line formed by starting with all open intervals and adding in all countable unions, countable intersections, and relative complements and continuing this process (by transfinite iteration through all countable ordinals) until the relevant closure properties are achieved (a construction ...
For example, {1, 2} is a subset of {1, 2, 3}, and so is {2} but {1, 4} is not. As implied by this definition, a set is a subset of itself. For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B, but A is not equal to B.
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.
By forming all subsets of a set S (all possible choices of its elements), we obtain the power set P(S). Georg Cantor proved that the power set is always larger than the set, i.e., |P(S)| > |S|. A special case of Cantor's theorem proves that the set of all real numbers R cannot be enumerated by natural numbers. R is uncountable: |R| > |N|.
Subsets: If x is a set, there exists a set y such that for each z, if , then . The import of III is that of Power Set above. Sketch of the proof of Power Set from III : for any class z that is a subclass of the set x , the class z is a member of the set y whose existence III asserts.