Chemistry Reference and Research
           
 
Periodic Table
- standard table
- large table
 
Chemical Elements
- by name
- by symbol
- by atomic number
 
Chemical Properties
 
Chemical Reactions
 
Organic Chemistry
 
Branches of Chemistry
Analytical chemistry
Biochemistry
Computational Chemistry
Electrochemistry
Environmental chemistry
Geochemistry
Inorganic chemistry
Materials science
Medicinal chemistry
Nuclear chemistry
Organic chemistry
Pharmacology
Physical chemistry
Polymer chemistry
Supramolecular Chemistry
Thermochemistry

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:

\forall A, \exists\; {\mathcal{P}A}, \forall B: B \in {\mathcal{P}A} \iff (\forall C: C \in B \implies C \in A)

Or in words:

Given any set A, there is a set PA such that, given any set B, B is a member of PA if and only if B is a subset of A.

By the axiom of extensionality this set 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.

01-04-2007 01:16:19
The contents of this article are licensed from Wikipedia.org under the GNU Free Documentation License. How to see transparent copy