How many elements are in the power set of a set with n elements?

The power set of a set is defined as the set of all possible subsets of that set, including the empty set and the set itself. For a set with ( n ) elements, each element can either be included in a subset or excluded from it. This binary choice for each of the ( n ) elements leads to a total of ( 2^n ) possible combinations.

To visualize this, consider a set with 3 elements, for example, ( {a, b, c} ). The subsets of this set include the following:

• The empty set: ( {} )

• Single-element subsets: ( {a}, {b}, {c} )

• Two-element subsets: ( {a, b}, {a, c}, {b, c} )

• The set itself: ( {a, b, c} )

Counting these, we find there are 8 subsets, which corresponds to ( 2^3 ). This pattern holds for any set—thus, the number of elements in the power set of a set with ( n ) elements is indeed ( 2^n \