Power Set
A power set includes all the subsets of a given set including the empty set. The power set is denoted by the notation P(S) and the number of elements of the power set is given by 2n. A set, in simple words, is a collection of distinct objects. If there are two sets A and B, then set A will be the subset of set B if all the elements of set A are present in set B. A power set can be imagined as a place holder of all the subsets of a given set, or, in other words, the subsets of a set are the members or elements of a power set.
Power Set Definition
A power set is defined as the set or group of all subsets for any given set, including the empty set, which is denoted by {}, or, ϕ. A set that has ‘n’ elements has 2n subsets in all. For example, let Set A = {1,2,3}, therefore, the total number of elements in the set is 3. Therefore, there are 23 elements in the power set. Let us find the power set of set A.
Set A = {1,2,3}
Subsets of set A = {}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}
Power set P(A) = { {}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3} }
Cardinality of a Power Set
The cardinality of a set is the total number of elements in the set. A power set contains the list of all the subsets of a set. The total number of subsets for a set of ‘n’ elements is given by 2n. Since the subsets of a set are the elements of a power set, the cardinality of a power set is given by |P(A)| = 2n. Here, n = the total number of elements in the given set.
Example: Set A = {1,2}; n = 2
|P(A)| = 2n = 22 = 4.
Subsets of A = {}, {1},{2},{1,2}
Therefore, |P(A)| = 4.