# Axiom of power set

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:

:$forall A , exists P , forall B , \left[B in P iff forall C , \left(C in B Rightarrow C in A\right)\right]$

where "P" stands for the power set, $mathcal\left\{P\right\}\left(A\right)$, of "A". In English, this says:

:Given any set "A", there is a set $mathcal\left\{P\right\}\left(A\right)$ such that, given any set "B", "B" is a member of $mathcal\left\{P\right\}\left(A\right)$ if and only if "B" is a subset of "A". (Subset is not used in the formal definition above because the axiom of power set is an axiom that may need to be stated without reference to the concept of subset.)

By the axiom of extensionality this set is unique.We call the set $mathcal\left\{P\right\}\left(A\right)$ the "power set" of "A". Thus, the essence of the axiom is that every set has a power set.

The axiom of power set is generally considered uncontroversial and it, or an equivalent axiom, appears in most alternative axiomatizations of set theory.

Consequences

The Power Set Axiom allows the definition of the Cartesian product of two sets $X$ and $Y$:

:$X imes Y = \left\{ \left(x, y\right) : x in X land y in Y \right\}.$

Notice that:$x, y in X cup Y$:$\left\{ x \right\}, \left\{ x, y \right\} in mathcal\left\{P\right\}\left(X cup Y\right)$:$\left(x, y\right) = \left\{ \left\{ x \right\}, \left\{ x, y \right\} \right\} in mathcal\left\{P\right\}\left(mathcal\left\{P\right\}\left(X cup Y\right)\right)$

and thus the Cartesian product is a set since

:$X imes Y subseteq mathcal\left\{P\right\}\left(mathcal\left\{P\right\}\left(X cup Y\right)\right).$

One may define the Cartesian product of any finite collection of sets recursively:

:$X_1 imes cdots imes X_n = \left(X_1 imes cdots imes X_\left\{n-1\right\}\right) imes X_n.$

Note that the existence of the Cartesian product can be proved in Kripke–Platek set theory which does not contain the power set axiom.

References

*Paul Halmos, "Naive set theory". Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).
*Jech, Thomas, 2003. "Set Theory: The Third Millennium Edition, Revised and Expanded". Springer. ISBN 3-540-44085-2.
*Kunen, Kenneth, 1980. "Set Theory: An Introduction to Independence Proofs". Elsevier. ISBN 0-444-86839-9.

----

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Power set — In mathematics, given a set S , the power set (or powerset) of S , written mathcal{P}(S), P ( S ), or 2 S , is the set of all subsets of S . In axiomatic set theory (as developed e.g. in the ZFC axioms), the existence of the power set of any set… …   Wikipedia

• set theory — the branch of mathematics that deals with relations between sets. [1940 45] * * * Branch of mathematics that deals with the properties of sets. It is most valuable as applied to other areas of mathematics, which borrow from and adapt its… …   Universalium

• Axiom of pairing — In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo Fraenkel set theory. Formal statement In the formal language of the Zermelo Frankel axioms, the …   Wikipedia

• Set theory — This article is about the branch of mathematics. For musical set theory, see Set theory (music). A Venn diagram illustrating the intersection of two sets. Set theory is the branch of mathematics that studies sets, which are collections of objects …   Wikipedia

• Axiom — This article is about logical propositions. For other uses, see Axiom (disambiguation). In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self evident or to define and… …   Wikipedia

• Axiom of choice — This article is about the mathematical concept. For the band named after it, see Axiom of Choice (band). In mathematics, the axiom of choice, or AC, is an axiom of set theory stating that for every family of nonempty sets there exists a family of …   Wikipedia

• Axiom schema of replacement — In set theory, the axiom schema of replacement is a schema of axioms in Zermelo Fraenkel set theory (ZFC) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite… …   Wikipedia

• Set of all sets — In set theory as usually formulated, referring to the set of all sets typically leads to a paradox. The reason for this is the form of Zermelo s axiom of separation: for anyformula varphi(x) and set A, the set {x in A mid varphi(x)}which contains …   Wikipedia

• Set (mathematics) — This article gives an introduction to what mathematicians call intuitive or naive set theory; for a more detailed account see Naive set theory. For a rigorous modern axiomatic treatment of sets, see Set theory. The intersection of two sets is… …   Wikipedia

• Axiom (computer algebra system) — Scratchpad redirects here. For scratchpad memory, see Scratchpad RAM. Axiom Developer(s) independent group of people Stable release September 2011 Operating system cross platform …   Wikipedia