Question

What is the power set of s={1,2,3,4,5}

Answer 1

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}}