The power set P(X) of a finite set X is the set of all possible subsets of X. If X has n elements, then it has a total of 2ⁿ subsets.
For the set S = {1, 2, 3, 4, 5}, we have
• subsets with cardinality 0:
{ } (the empty set)
• subsets with cardinality 1:
{1}, {2}, {3}, {4}, {5}
• subsets with card. 2:
{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 4}, {4, 5}
• subsets with card. 3:
{1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 3, 4}, {1, 3, 5}, {1, 4, 5}, {2, 3, 4}, {2, 3, 5}, {2, 4, 5}, {3, 4, 5}
• subsets with card. 4:
{1, 2, 3, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {1, 3, 4, 5}, {2, 3, 4, 5}
• subsets with card. 5:
{1, 2, 3, 4, 5}
Then P(S) is the set containing all of these sets,
{{}, {1}, {2}, {3}, …, {1, 2, 3, 4, 5}}