Power set

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Given a set S, the power set of S, written P(S), 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 sets that are also subsets of S as well; the complete list is as follows:

  • {}
  • {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} }