Encyclopedia > Axiom of power set

  Article Content

Axiom of power set

In mathematics, the axiom of power set is one of the Zermelo-Fraenkel axioms of axiomatic set theory.

In the formal language of the Zermelo-Fraenkel axioms, the axiom reads:

A, ∃ B, ∀ C, CB ↔ (∀ D, DCDA);
or in words:
Given any set A, there is a set B such that, given any set C, C is a member of B if and only if, given any set D, if D is a member of C, then D is a member of A.

To understand this axiom, note that the clause in parentheses in the symbolic statement above simply states that C is a subset of A. Thus, what the axiom is really saying is that, given a set A, we can find a set B whose members are precisely the subsets of A. We can use the axiom of extension to show that this set B is unique. We call the set B the power set of A, and denote it PA. Thus the essence of the axiom is:

Every set has a power set.

The axiom of power set is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatisation of set theory.

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
French resistance

... Many groups were not enthusiastic at first. When Germans initiated a forced labor draft in France in the beginning of 1943, thousands of young men fled and joined ...

This page was created in 74.6 ms