Power set

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 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} }

In case you were wondering, for any given set S, S is always considered to be a subset of itself (by definition) - a proper subset is any subset except the set itself. Also, the empty set, written {}, is also a subset of any given set S. This is because the empty set vacuously satisfies the definition of a subset of S; since the empty set has no elements, every element it the empty set is also an element of S, for any given set S.

If the above paragraph sounds a bit like double-speak - it is only because it is stating the obvious.