Power Set

The power set of a set contains all its subsets, including the empty set and itself. If A is a set, its power set is denoted by 𝒫(A).

Examples

Let A = {1,2}. Then the subsets are:

• ∅ (empty set)
• {1}
• {2}
• {1,2}

Thus, the power set: 𝒫(A) = { ∅, {1}, {2}, {1,2} }.

In general, if |A| = n, then the number of elements in the power set is 2^n. This is because each element either appears in a subset or not.

Properties

• Every power set contains the empty set and itself.
• |𝒫(A)| = 2^|A|, where |A| is the number of elements in A.
• If A is a subset of B, then 𝒫(A) is a subset of 𝒫(B).

Everyday Example

If you have two shirts in a closet (red and blue), the possible 'shirt choices' are subsets: wear nothing, only the red, only the blue, or both. These together form the power set.

Summary

The power set helps organize what combinations are possible from the elements of a given set. This concept is key in combinatorics and computer science (e.g., examining all possible states).

Practice Exercise