Power set

In mathematics, given a set S, the power set (or powerset) of S, written mathcal{P}(S) or 2S, 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 is postulated by the axiom of power set.

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Any subset F of mathcal{P}(S) is called a family of sets over S.

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For example, if S is the set {A, B, C} then the complete list of subsets of S is as follows:

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• {} (the empty set)
• {A}
• {B}
• {C}
• {A, B}
• {A, C}
• {B, C}
• {A, B, C}

and hence the power set of S is

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:mathcal{P}(S) = {{}, {A}, {B}, {C}, {A, B}, {A, C}, {B, C}, {A, B, C}},!

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If S is a finite set with |S|=n elements, then the power set of S contains |mathcal{P}(S)| = 2^n elements. (One can - and computers actually do - represent the elements of mathcal{P}(S) as n-bit numbers; the n-th bit refers to presence or absence of the n-th element of S. There are 2n such numbers.)

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One can also consider the power set of infinite sets. Cantor's diagonal argument shows that the power set of a set (infinite or not) always has strictly higher cardinality than the set itself (informally the power set must be 'greater' than the original set). The power set of the set of natural numbers for instance can be put in a one-to-one correspondence with the set of real numbers (by identifying an infinite 0-1 sequence with the set of indices where the ones occur).

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The power set of a set S, together with the operations of union, intersection and complement forms the prototypical example of a boolean algebra. In fact, one can show that any finite boolean algebra is isomorphic to the boolean algebra of the power set of a finite set. For infinite boolean algebras this is no longer true, but every infinite boolean algebra is a subalgebra of a power set boolean algebra.

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The power set of a set S forms an Abelian group when considered with the operation of symmetric difference (with the empty set as its unit and each set being its own inverse) and a commutative semigroup when considered with the operation of intersection. It can hence be shown (by proving the distributive laws) that the power set considered together with both of these operations forms a commutative ring.

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Power set: The notation 2^S ►



 

Set: :This article is about sets in mathematics. For other senses, see set (disambiguation)....

Subset: In mathematics, especially in set theory, a set A is a subset of a set B, if A is "contained" inside B. The relationship of one set being a subset of another is called inclusion....

Axiomatic set theory: Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions mad...




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