The power set of S is the set of all its subsets, including the empty set and S itself. Power sets appear in probability (events), logic, computer science (bit masks) and combinatorics.
How to use the Power Set Calculator
- Set elements (separate with commas)
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Each element is either in or out of a subset, so n elements give 2ⁿ subsets. The calculator generates them with binary counting and sorts them by size.
Formula
|P(S)| = 2ⁿ, number of k-element subsets = C(n, k)
Worked example
Inputs — Set elements (separate with commas): a, b, c.
Results — Number of subsets = 8; All subsets = ∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}; Subsets by size = 0 elements: 1
1 element: 3
2 elements: 3
3 elements: 1; Number of elements = 3.
Key terms
- Power set
- The set of all subsets of a set.
- Cardinality
- The number of elements in a set.
Frequently asked questions
How many subsets does a set with 4 elements have?
2⁴ = 16.
Is the empty set a subset?
Yes, of every set.
What is the power set of the empty set?
{∅} – it has one element.
Why 2ⁿ?
Each of the n elements can independently be in or out of a subset.