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Power set

Given a set S, the power set of S, written P(S) or 2S, is the set of all subsets of S. The definition of a subset is as follows: Given sets S and T, then T is defined to be a subset of S if every element of T is also an element of S.

If S is the set {A, B, C} then {A,C} is a subset of S. There are other subsets of S; the complete list is as follows:

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

So the power set of S, written P(S), is the set containing all the subsets above. Written out this would be the set:

P(S) = { {}, {A}, {B}, {C}, {A, B}, {A, C}, {B, C}, {A, B, C} }

If n = |S| is the number of elements of S, then the respective power set contains |P(S)| = 2n elements. (One can [and computers actually do] represent the elements of 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.)

One can also consider the power set of infinite sets. Cantor's diagonal argument shows that the power set of an infinite set always has strictly higher cardinality than the set itself (informally the power set must be 'more infinite' than the original set). The power set of the 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).

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 S. 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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