Power set

In mathematics, the power set (or powerset) of any set S, written P(S), ℘(S), P(S), ℙ(S) or 2^S, is the set of all subsets of S, including the empty set and S itself. 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.

Any subset of P(S) is called a family of sets over S.

Example

If S is the set {x, y, z }, then the subsets of S are:

{} (also denoted or , the empty set or the null set)

and hence the power set of S is {{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}.

Properties

If S is a finite set with |S| = n elements, then the number of subsets of S is |P(S)| = 2ⁿ. This fact, which is the motivation for the notation 2^S, may be demonstrated simply as follows,

Cantor's diagonal argument shows that the power set of a set (whether infinite or not) always has strictly higher cardinality than the set itself (informally the power set must be larger than the original set). In particular, Cantor's theorem shows that the power set of a countably infinite set is uncountably infinite. For example, the power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers (see: cardinality of the continuum).

Powerset (company)

Powerset is an American company based in San Francisco, California, that, in 2006, was developing a natural language search engine for the Internet. It became a wholly owned by American software and computing giant Microsoft.

Powerset was working on building a natural language search engine that could find targeted answers to user questions (as opposed to keyword based search). For example, when confronted with a question like "Which U.S. state has the highest income tax?", conventional search engines ignore the question phrasing and instead do a search on the keywords "state", "highest", "income", and "tax". Powerset on the other hand, attempts to use natural language processing to understand the nature of the question and return pages containing the answer.

The company was in the process of "building a natural language search engine that reads and understands every sentence on the Web". The company has licensed natural language technology from PARC, the former Xerox Palo Alto Research Center.