Enter up to three sets (A, B, C) and a universal set (U) using comma-separated values, then choose your set operation — Union, Intersection, Complement, Difference, or Power Set. The Set Theory Calculator returns the resulting set, its cardinality, and a visual breakdown of which elements appear in each region.
Result Set
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Cardinality (Number of Elements)
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Elements Only in A
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Elements Only in B
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Elements in Both A and B
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The union of sets A and B (written A ∪ B) is the set of all elements that belong to A, to B, or to both. Duplicate elements are included only once. For example, if A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}.
The intersection (A ∩ B) contains only the elements that appear in both A and B simultaneously. Using the example A = {1, 2, 3} and B = {3, 4, 5}, the intersection is {3}. If no elements are shared, the result is the empty set ∅.
The complement of set A (Aᶜ) contains all elements in the universal set U that are not in A. You must define U for this operation. For instance, if U = {1, 2, 3, 4, 5} and A = {1, 2}, then Aᶜ = {3, 4, 5}.
The set difference A − B contains elements that are in A but not in B. It is not the same as B − A. For example, if A = {1, 2, 3, 4} and B = {3, 4, 5}, then A − B = {1, 2} and B − A = {5}.
The symmetric difference (A △ B) contains elements that are in A or B but not in both — essentially the union minus the intersection. For A = {1, 2, 3} and B = {3, 4, 5}, the symmetric difference is {1, 2, 4, 5}.
The power set of A is the collection of all possible subsets of A, including the empty set and A itself. If A has n elements, the power set has 2ⁿ subsets. For example, if A = {1, 2}, the power set is {∅, {1}, {2}, {1, 2}}. Note: large sets produce very large power sets.
Yes. Elements are treated as strings, so you can enter letters, words, or numbers. Just separate them with commas — for example, 'a, b, c' or 'apple, banana, cherry'. Whitespace around commas is ignored automatically.
Cardinality refers to the number of elements in a set. A set with 5 elements has cardinality 5, written |A| = 5. The cardinality of the empty set is 0. This calculator displays the cardinality of the result set after every operation.