Set Calculator: Union, Intersection, Difference, Complement and Power Set

How it works

A ∪ B = {x : x ∈ A or x ∈ B} · A ∩ B = {x : x ∈ A and x ∈ B} · A − B = {x : x ∈ A and x ∉ B} · A Δ B = (A − B) ∪ (B − A) · A′ = U − A · n(A ∪ B) = n(A) + n(B) − n(A ∩ B) · power set size = 2ⁿ⁽ᴬ⁾ · n(A × B) = n(A) × n(B)

Type each set as a list of elements separated by commas; braces are optional, numbers and words both work, and a repeated element counts once. The calculator returns every basic operation at once: the union (elements in A or B), the intersection (in both), the two differences A − B and B − A, the symmetric difference (in exactly one of the sets), and, if you give a universal set U, the complements A′ and B′. Pick the one you need as the main result. It also counts the elements, checks the inclusion–exclusion rule n(A ∪ B) = n(A) + n(B) − n(A ∩ B), says whether one set is a subset of the other or whether they are disjoint, lists the regions of the Venn diagram (for two or three sets), lists the power set of A grouped by size when A has up to 8 elements, and writes out the Cartesian product A × B when it has up to 100 pairs. Choose “My own expression” to evaluate any combination such as (A ∪ B) ∩ C′ or A − (B ∩ C). Intersection is applied before union, difference and symmetric difference, which are applied left to right; use parentheses to be explicit. The worked steps show the result of every sub-expression, which makes the page useful for checking De Morgan’s laws by hand. The probability-of-union calculator works with probabilities; this one works with the actual elements.

Worked example

With A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7} and U = {1, …, 10}: A ∪ B = {1, 2, 3, 4, 5, 6, 7}, A ∩ B = {4, 5}, A − B = {1, 2, 3}, B − A = {6, 7}, A Δ B = {1, 2, 3, 6, 7}, A′ = {6, 7, 8, 9, 10} and B′ = {1, 2, 3, 8, 9, 10}. The count check is 5 + 4 − 2 = 7. A and B overlap without either containing the other, A has 2⁵ = 32 subsets, and A × B has 5 × 4 = 20 ordered pairs. The expression (A ∪ B)′ gives {8, 9, 10}, the same as A′ ∩ B′.

Sources: Pearson: Set operations calculator (union, complement, power set and Cartesian examples) · Math is Fun: Set calculator · Vedantu: Set calculator (union, intersection, difference examples)

Frequently asked questions