Power set
P(S) = {A | A ⊆ S}
The power set is the set whose elements are all subsets of S.
Power Set Calculator
Enter a set
Separate elements with commas, semicolons, or new lines. Optional outer braces are accepted. Elements are case-sensitive text labels.
Private by design: your set is parsed and generated on this device. Inputs are not uploaded, saved, or added to the page URL.
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Power set formula and definitions
Number of subsets
|P(S)| = 2ⁿ, where n = |S|
Each element independently has two choices: included or excluded.
Subsets of size k
count = C(n, k) = n! / (k!(n − k)!)
The number of k-element subsets is the binomial coefficient “n choose k.”
Proper subsets
proper subsets = 2ⁿ − 1
Every subset except S itself is proper. This count includes ∅ when S is nonempty.
Adding the binomial counts across all possible sizes gives Σ C(n, k) = 2ⁿ. The calculator’s “group by subset size” order follows these levels from k = 0 through k = n.
Power set examples
Empty set
P(∅) = {∅}
The empty set has exactly one subset: itself.
Two elements
P({a, b}) = {∅, {a}, {b}, {a, b}}
Two elements produce 2² = 4 subsets.
Three elements
P({a, b, c})
There is 1 subset of size 0, with 3 of size 1, 3 of size 2, and 1 of size 3: 1 + 3 + 3 + 1 = 8.
Exponential growth, method, and limits
The calculator assigns one bit to each distinct input element. A 1 includes that element and a 0 excludes it, so the integers from 0 through 2ⁿ − 1 represent every subset exactly once. “Binary-mask order” displays that direct enumeration; “group by subset size” sorts the same masks by their number of included elements.
| Elements n | Subsets 2ⁿ | Proper subsets | Subsets of size 2 |
|---|---|---|---|
| 5 | 32 | 31 | 10 |
| 10 | 1,024 | 1,023 | 45 |
| 15 | 32,768 | 32,767 | 105 |
| 20 | 1,048,576 | 1,048,575 | 190 |
Generation is limited to 15 unique elements because the output doubles with each added element. Results are paginated so only the selected page is added to the document, while copy and CSV export include every generated subset. Input is limited to 2,000 characters, and individual labels may contain at most 80 characters. Commas, semicolons, and line breaks cannot be part of an element.
Repeated values are collapsed to one because a set contains unique elements. Labels are compared exactly after Unicode normalization: A and a remain different. Listing order is for readability only; mathematical sets themselves are unordered.
Power set calculator FAQ
What is a power set?
The power set P(S) contains every subset of S, from the empty set through S itself.
How many subsets does a set have?
A set with n distinct elements has 2ⁿ subsets. Each element can be either included in or excluded from a subset.
Is the empty set included in every power set?
Yes. The empty set ∅ is a subset of every set. It appears as the single size-0 subset.
What is the power set of the empty set?
P(∅) = {∅}. Be careful to distinguish ∅, which has zero elements, from {∅}, which has one element.
What is the difference between a subset and a proper subset?
A subset may equal the original set. A proper subset cannot equal it, so only the original set is removed from the full power set when counting proper subsets.
Why are duplicate input elements removed?
A mathematical set cannot contain repeated elements. For example, a, a, b represents the same set as a, b and therefore has four subsets, not eight.
Why is there a 15-element limit?
The number of results grows exponentially. Fifteen unique elements already create 32,768 subsets; a larger unrestricted input could make the page or exported file unresponsive.
Does subset order matter?
No. A set is unordered. Input order and the selected listing order only control how results are displayed and exported.
Does the calculator store my set?
No. Parsing, generation, filtering, copying, and CSV creation run locally in your browser. The tool does not transmit or save entered values.