The power set P(S) of a set S is the set of all possible subsets of S, including the empty set ∅ and S itself. For example, if S = {a, b}, then P(S) = {∅, {a}, {b}, {a, b}}.

How many subsets does a set with n elements have?

Exactly 2ⁿ subsets. Each element is either included or excluded, giving two independent choices per element. A set with 3 elements has 8 subsets; with 10 elements it has 1,024 subsets.

Is the empty set always included?

Yes. The empty set ∅ is a subset of every set, so it always appears in the power set. It represents choosing none of the elements.

Why is there a 20-element limit?

A set with 20 elements produces 2²⁰ = 1,048,576 subsets. Larger sets would generate millions of lines that freeze your browser. Use the formula 2ⁿ to compute the count for larger sets.

Are duplicate elements removed?

Yes. A set by definition contains unique elements. The tool silently deduplicates your input before computing, so {a, b, a} is treated as {a, b}.

What notation does the output use?

Each subset is shown with curly-brace notation, e.g. {a, b}. The empty set is displayed as ∅. Subsets are grouped by size for easy scanning.

Power Set Generator

Enter a set of elements and instantly see every possible subset, organized by size, with a total count.

Input Set

Elements (comma-separated)

Set: {a, b, c} — 3 elements → 8 subsets

Group by subset size Include empty set (∅)

Power Set 8 subsets

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Summary

The power set of a set S is the collection of ALL subsets of S, including the empty set and S itself. For a set with n elements, the power set contains exactly 2ⁿ subsets. This tool enumerates every subset, groups them by cardinality (size), and displays a total count so you can verify combinatorial problems at a glance.

How it works

Enter your elements in the input box, separated by commas (e.g. a, b, c).
Duplicate elements are automatically removed to keep the set valid.
The tool computes all 2ⁿ subsets using a bitmask enumeration algorithm.
Subsets are grouped by size — from the empty set (size 0) up to the full set (size n).
Use Copy to copy all subsets to your clipboard, or Download to save as a text file.

Use cases

Enumerate all possible subsets for a combinatorics homework problem.
Verify power set size (2ⁿ) when studying set theory.
Generate all feature flag combinations for software configuration testing.
List every possible sub-team from a small group of people.
Explore all binary attribute combinations in a decision tree or logic problem.
Check all possible hand sizes in a card game scenario.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu