Power Set
Generate the power set of any set with all possible subsets listed, including empty set. Free online set theory calculator for combinatorics and discrete math.
About This Calculator
The Power Set Calculator generates the complete power set (set of all subsets) for any set you enter. Simply type the elements as a comma-separated list, and the calculator produces every possible subset from the empty set to the full set. The total number of subsets (2^n where n is the number of elements) is displayed along with a numbered breakdown of each subset.
Power sets are a fundamental concept in set theory and combinatorics. The number 2^n arises because each of the n elements can be independently chosen to be included or excluded from any given subset. The power set always includes the empty set (empty) and the original set itself. This concept has applications in probability (enumerating all possible outcomes), computer science (binary decision trees, Boolean algebra), and mathematics (relations, functions, and topology).
This tool is designed for students studying set theory, combinatorics, and discrete mathematics, as well as teachers and professionals who need to enumerate subsets for analysis. Each subset is displayed with proper set notation, making it easy to copy and use in assignments, proofs, or further analysis.
Frequently Asked Questions
What is a power set?
The power set of a set S is the set of all possible subsets of S, including the empty set and S itself. If S has n elements, the power set has 2^n elements. For example, if S = {a, b, c}, the power set has 2^3 = 8 subsets: empty, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}.
How do I use the power set calculator?
Enter the elements of your set as a comma-separated list in the input field. For example, type 'a, b, c' and click Generate Power Set. The calculator displays the total number of subsets (2^n) and lists every subset from the empty set to the full set in a clear numbered breakdown table.
Why is the power set size 2^n?
For each element in the original set, there are two choices: either include it in a subset or exclude it. With n independent choices, the total number of possible combinations is 2 x 2 x ... x 2 (n times) = 2^n. This includes the empty set (all excluded) and the full set (all included).
What is an example of a power set?
For the set {1, 2}, the power set has 2^2 = 4 subsets: the empty set empty, {1}, {2}, and {1, 2}. For the set {a, b, c, d} (4 elements), the power set has 2^4 = 16 subsets. The power set always contains the empty set and the original set itself as subsets.
Is this power set tool free?
Yes, all calculators on Calculy are completely free to use with no registration or hidden charges. You can bookmark or share your power set results via the URL.
What is the power set used for?
Power sets are used in combinatorics for counting subsets, in probability for defining sample spaces, in computer science for algorithm analysis and Boolean algebra, in mathematics for defining functions and relations, and in data science for feature selection and subset generation.