Power Set Calculator
The power set of a set is the set of all of its subsets, from the empty set at one extreme to the whole set itself at the other, and everything in between. It is a fundamental object in set theory, combinatorics and computer science, where it underlies counting arguments and the design of search spaces. A set with n distinct elements has exactly two to the power n subsets, because each element is independently either in or out of a given subset, giving two choices per element. This calculator takes a finite set entered as comma-separated elements, removes any duplicates, and lists every subset along with the total count. It builds the subsets by treating the numbers from zero up to two to the n minus one as binary masks, where each bit decides whether the matching element is included, which is the standard and exhaustive way to enumerate a power set. The result is complete: no subset is skipped and none is repeated. Every figure is computed deterministically from the elements you enter, never sampled or truncated, so the same set always produces the same power set. The method and a worked example that reconciles to the calculator default are shown in full below.
A set with n elements has 2^n subsets. The default set {a, b, c} has 8 subsets: the empty set, {a}, {b}, {c}, {a, b}, {a, c}, {b, c} and {a, b, c}.
Formula
Number of subsets = 2^n, where n is the count of distinct elements
Each subset corresponds to a binary mask over the elements
The empty set and the full set are always included
The distinct elements are numbered, then the integers from 0 to 2^n minus 1 are read as binary masks. A 1 in a bit position includes the matching element, so iterating over all masks lists every subset exactly once.
Worked example
Find the power set of the set {a, b, c}, which has three distinct elements.
- n = 3, so the number of subsets is 2^3 = 8
- Masks 000 to 111 give: {}, {a}, {b}, {a,b}, {c}, {a,c}, {b,c}, {a,b,c}
- That is 8 subsets, including the empty set and the full set
This is the calculator's default set, so the 8 subsets above match the widget exactly.
Power Set Calculator: frequently asked questions
What is a power set?
The power set of a set is the collection of all of its subsets. It always contains the empty set and the original set itself, plus every subset in between.
How many subsets does a set have?
A set with n distinct elements has exactly two to the power n subsets, because each element is independently either included or excluded, giving two choices per element.
Is the empty set always included?
Yes. The empty set is a subset of every set, so it is always a member of the power set, as is the set itself.
Why is there a limit on the number of elements?
The number of subsets doubles with each element, so even a dozen elements produce thousands of subsets. The calculator caps the input to keep the listing readable.
Are duplicate elements counted twice?
No. The calculator removes duplicates first, since a set contains each distinct element only once. The count uses the number of distinct elements.
Official sources
- Set theory and combinatorics reference: US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.