Power Set Calculator

The power set of a set S is the collection of all possible subsets of S, including the empty set and S itself. If S has n elements, its power set has exactly 2^n subsets.

That exponential growth is dramatic. A set with 3 elements has 8 subsets. A set with 10 elements has 1024. A set with 20 elements has over one million. This is why this calculator caps input at 15 elements: 2^15 = 32,768 subsets, which is the practical limit for listing them all in a browser.

The power set is denoted P(S) or 2^S. The notation 2^S reflects exactly the fact that each element independently has two choices: either it is in a subset or it is not. Since there are n elements and two choices per element, the total number of subsets is 2 * 2 * … * 2 (n times) = 2^n.

The empty set is always a subset (0 elements chosen). The full set S is always a subset (all n elements chosen). Everything else is somewhere in between.

Power sets appear in combinatorics and discrete mathematics whenever you need to consider all possible combinations. They are the formal foundation for the binomial theorem: the number of k-element subsets is C(n,k), and summing C(n,k) for k from 0 to n gives 2^n.

In database theory, the power set of attributes is used to enumerate all possible functional dependencies. In logic, the power set of a set of propositions corresponds to all possible truth assignments. In computer science, the power set construction is used to convert nondeterministic finite automata (NFAs) into deterministic ones (DFAs).

The subsets are listed in binary order: the empty set first, then singletons, then pairs, and so on. Each subset corresponds to a binary number from 0 to 2^n - 1, where a 1 in position i means element i is included.