Math
Power Set Calculator
The power set P(A) of a set A is the set of all its subsets, including the empty set ∅ and A itself.
Type the elements and the power set calculator lists every subset grouped by size, counts them with 2^n, and shows how the binomial coefficients C(n, k) add up to that total.
Separate elements with commas or spaces. Braces are optional. Up to 6 elements.
Try an example
Result
Power set
—
- Number of subsets (2^n)
- —
- Elements in the set (n)
- —
- Subsets grouped by size
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- Proper subsets (2^n − 1)
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- Set as read
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Study path
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What the power set calculator solves
Every element of a set is either in a given subset or not, two choices each, so n elements produce 2^n subsets. Listing them by size reproduces a row of Pascal's triangle: C(n, 0) subsets with no elements, C(n, 1) with one, and so on up to the single subset with all n.
| n | Subsets (2^n) | Proper subsets (2^n − 1) |
|---|---|---|
| 0 | 1 | 0 |
| 1 | 2 | 1 |
| 2 | 4 | 3 |
| 3 | 8 | 7 |
| 4 | 16 | 15 |
| 5 | 32 | 31 |
| 6 | 64 | 63 |
How to use it
- Type the elements separated by commas or spaces, with or without braces.
- Read the full power set and the count 2^n.
- Use the grouped-by-size line to see the C(n, k) breakdown for a specific subset size.
How to read the answer
The power set contains 2^n subsets: one empty subset, n singletons, C(n, 2) pairs, and so on. A proper subset is any subset other than the set itself, so there are 2^n − 1 of them. Duplicated elements are counted once.
Common mistakes and edge cases
- Forgetting the empty set and the set itself; both are subsets.
- Listing {a, b} and {b, a} as different subsets; order does not matter in a set.
- Counting n^2 instead of 2^n subsets.
- Confusing 'proper subsets' (2^n − 1) with 'non-empty subsets' (also 2^n − 1, but a different collection).
Worked examples
Three elements
{a, b, c} has 8 subsets
Power set
{∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}
Numbers
{1, 2} has 4 subsets
Power set
{∅, {1}, {2}, {1, 2}}
Single element
{x} has 2 subsets: ∅ and {x}
Power set
{∅, {x}}
Empty set
∅ has exactly one subset
Power set
{∅}
Duplicates are ignored
{a, a, b} is the set {a, b}
Power set
{∅, {a}, {b}, {a, b}}
Too many elements
Seven elements would give 128 subsets
Power set
Error
Frequently asked questions
What is a power set?+
The set of all subsets of a set, including the empty set and the set itself. For A = {1, 2}, P(A) = {∅, {1}, {2}, {1, 2}}.
How many subsets does a set with n elements have?+
2^n. Each element is either included or excluded, and the choices multiply.
Is the empty set in every power set?+
Yes. ∅ is a subset of every set, so it is always a member of the power set. The power set of ∅ itself is {∅}, which has one element.
What is the difference between a subset and a proper subset?+
A proper subset leaves out at least one element. Every set has 2^n subsets but only 2^n − 1 proper subsets.
How many subsets have exactly k elements?+
C(n, k), the binomial coefficient. For a 4-element set there are C(4, 2) = 6 two-element subsets.
Why does the calculator stop at 6 elements?+
The list doubles with each element: 7 elements give 128 subsets and 10 give 1,024. Use the formula 2^n for larger sets.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
References
- OpenStax, Contemporary Mathematics, Section 1.2 “Subsets” — Subsets, proper subsets, and counting subsets with 2^n.
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Last updated: September 4, 2026