How to use the power set calculator
A power set is the collection of every subset that can be formed from a set. This calculator lists the complete power set for up to 10 distinct text elements, includes the empty set, and groups subsets by their number of elements so a long result remains traceable.
- Enter up to 10 comma-separated elements. A completely blank field is accepted as the empty set.
- Review capitalization and separators. Outer spaces and duplicate entries are removed before the element limit and subset count are applied.
- Select Generate power set. The result first reports 2ⁿ, then shows groups for zero-element, one-element, and larger subsets through the full original set.
- Use Copy result to copy the currently confirmed grouped list. Editing or clearing the input immediately removes the old list and disables copying.
Why a set with n elements has 2ⁿ subsets
For each distinct element there are two independent choices: include it in a subset or leave it out. Multiplying two choices across n elements gives 2 × 2 × … × 2 = 2ⁿ possible subsets. This total includes both improper subsets: the empty set ∅ and the original set itself.
Grouping by size gives binomial counts. A set of n elements has C(n, k) subsets containing exactly k elements, and C(n, 0) + C(n, 1) + … + C(n, n) = 2ⁿ. The page enumerates subsets directly, then groups them by k; it does not merely display the formula.
Worked examples
Three elements produce eight subsets
For {a, b, c}, the zero-element group is ∅. The one-element group is {a}, {b}, {c}; the two-element group is {a, b}, {a, c}, {b, c}; and the three-element group is {a, b, c}. The group sizes 1 + 3 + 3 + 1 total 8 = 2³.
Duplicates and the empty set
Entering red, red, blue first becomes {red, blue}, so the result has 2² = 4 subsets rather than eight. A blank input is ∅; its power set contains one member, the empty subset, so |P(∅)| = 2⁰ = 1.
Enumeration limits and interpretation
- After trimming and duplicate removal, the input may contain no more than 10 distinct elements. Ten elements already produce 1,024 subsets, so larger inputs are rejected before enumeration begins.
- Each element may contain at most 40 characters. Commas are separators and cannot be part of an element. Nested sets, quoted comma values, and mathematical expressions are outside this text-based parser.
- Uppercase and lowercase are different. Input order is used to keep the generated list predictable, although subset membership itself has no mathematical order.
- The list is complete within the limit, not a random sample. The calculator does not attempt unbounded generation, symbolic set simplification, or streaming output.
Frequently asked questions
Is the empty set included?
Yes. It is the only subset in the zero-element group and belongs to every power set.
Why are duplicate inputs removed?
A set records membership, not repetition. Removing duplicates gives the correct value of n before calculating 2ⁿ.
Why is the maximum only 10 elements?
Enumeration doubles with every additional element. The fixed limit prevents an unexpectedly huge page and stops before calculation.
What order are the subsets shown in?
Groups run from zero elements through n elements. Within each group, subsets follow the first-seen order of input elements.
Can I use the result as a combination list?
Yes, when you need every possible selection including choosing none and choosing all. Read the group for a particular size if you only need k-element combinations.