Power Set Calculator
This power set calculator lists the subsets of your set. Example: the set {a, b, c} has 3 elements, so its power set has 2³ = 8 subsets, including ∅ and {a, b, c}.
Source: Wikipedia – Power set. Updated: .
How it is calculated
The power set of a set M is the set of all its subsets, including the empty set ∅ and M itself. A set with n elements has exactly 2ⁿ subsets.
Why 2ⁿ?
Each element can be in a subset or not, which is two choices. With n elements these choices multiply to 2 · 2 · … · 2 = 2ⁿ. If you sort the subsets by their size k, there are exactly C(n, k) of each size, and the sum of all C(n, k) is again 2ⁿ.
Worked examples
For M = {a, b, c}, n = 3 and the power set has 2³ = 8 elements: ∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c} and {a, b, c}. By size they split into 1, 3, 3, 1, since C(3, 0) = 1, C(3, 1) = 3, C(3, 2) = 3 and C(3, 3) = 1. Four elements give 2⁴ = 16 subsets (1, 4, 6, 4, 1), ten elements already 1024. The empty set has exactly one subset, namely itself.
- The order of elements does not matter; {a, b} and {b, a} are the same set.
- The count grows very fast: 20 elements give over a million subsets.
- The power set is written P(M) or 2^M.
Frequently asked questions
What is a power set?
The set of all subsets of a set, including ∅ and the set itself. For {a, b, c} that is 8 subsets.
How many elements does the power set have?
2ⁿ for a set with n elements. 4 elements give 16 subsets, 10 elements give 1024.
Is the empty set in the power set?
Yes, ∅ is a subset of every set and so always an element of the power set.
Sources and legal basis
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