Power Set

Power set of a set S is the set of all subsets of S including the empty set. The cardinality of a power set of a set S of cardinality n is 2ⁿ. Power set is denoted as P(S).

Example −

For a set S = { a, b, c, d } let us calculate the subsets −

• Subsets with 0 elements − { ∅ } (the empty set)

• Subsets with 1 element − { a }, { b }, { c }, { d }

• Subsets with 2 elements − { a, b }, { a,c }, { a, d }, { b, c }, { b,d },{ c,d }

• Subsets with 3 elements − { a ,b, c},{ a, b, d }, { a,c,d },{ b,c,d }

• Subsets with 4 elements − { a, b, c, d }

Hence, P(S)=

{ { ∅ }, { a }, { b }, { c }, { d }, { a,b }, { a,c }, { a,d }, { b,c }, { b,d }, { c,d }, { a,b,c }, { a,b,d }, { a,c,d }, { b,c,d }, { a,b,c,d } }

| P(S) | = 2⁴ = 16

Note − The power set of an empty set is also an empty set.

| P { ∅ } | = 2⁰ = 1