Enter your set elements (comma-separated) into the Power Set Calculator and get back every possible subset of that set. You'll see the complete power set listed out, along with the cardinality (total number of subsets) — so you can verify your work or explore combinatorics without tedious manual listing.
Number of Subsets (Cardinality)
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Size of Original Set (n)
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Power Set (All Subsets)
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A power set is the set of all possible subsets of a given set, including the empty set and the set itself. For example, if your set is {1, 2}, its power set is { {}, {1}, {2}, {1,2} }. Every set has a power set, no matter how large or small.
The number of subsets (the cardinality of the power set) is 2ⁿ, where n is the number of elements in the original set. So a set with 3 elements has 2³ = 8 subsets. A set with 5 elements has 2⁵ = 32 subsets.
A set with 4 elements has 2⁴ = 16 subsets in its power set. These range from the empty set {} all the way up to the full set itself, covering every possible combination of the 4 elements.
The power set of an empty set contains exactly one element: the empty set itself. So P({}) = { {} }. Its cardinality is 2⁰ = 1.
The power set of a set A is typically written as P(A) or ℙ(A). Each subset is listed inside curly braces. For example, if A = {a, b}, then P(A) = { {}, {a}, {b}, {a,b} }.
No, a power set can never be empty. Even the power set of the empty set contains one element (the empty set itself). The minimum cardinality of any power set is 1.
For each element in your set, you can either include it or exclude it — that's 2 choices per element. List every combination systematically: start with the empty set, then add each individual element, then all pairs, then triples, and so on up to the full set. This gets tedious quickly, which is why a calculator helps.
Yes. The power set includes all subsets, both proper and improper. The full original set is an improper subset of itself and is always included. The empty set is also always included as a subset.