Power Set

The power set of a set S is the set of all subsets of S, including the empty set and S itself. The power set is denoted as P(S). If S has n elements, then its power set has 2ⁿ elements.

Example

For a set S = { a, b, c, d }, let us list all the subsets grouped by size −

Subsets with 0 elements:  { ? }
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 }

The complete power set is −

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)| = 2? = 16

The count of subsets at each size follows the binomial coefficients −

Power Set of the Empty Set

The power set of the empty set contains exactly one element − the empty set itself −

P(?) = { ? }

|P(?)| = 2? = 1

Note − The power set of an empty set is not empty. It contains one element (the empty set), so its cardinality is 1.

Power Set Cardinality for Small Sets

Conclusion

The power set P(S) contains all possible subsets of S, including the empty set and S itself. For a set with n elements, the power set always has exactly 2ⁿ elements.