Set Operations Calculator (Union & Intersection)
Enter two sets of elements and instantly compute their union, intersection, differences, symmetric difference and all cardinalities.
Operation
Number of elements in the result set
7
totalA only
42.9%
A ∩ B
28.6%
B only
28.6%
- 1
|A|
5 - 2
|B|
4 - 3
|A ∩ B| (shared elements)
2 - 4
|A ∪ B| = |A| + |B| − |A ∩ B|
5 + 4 − 2 = 7Inclusion–exclusion: subtract the overlap to avoid counting it twice.
How does this calculator work?
Type two sets and pick an operation. The tool removes duplicates, then returns the union, intersection, one-sided differences or symmetric difference along with every cardinality. A Venn-style donut splits the union into A-only, shared and B-only elements so you can see exactly how the two sets overlap.
Formula
How this is calculated
You type the members of Set A and Set B into two text boxes. Tokens may be separated by commas, spaces or newlines, and any repeated token is collapsed so each set contains only unique elements — exactly matching the mathematical definition of a set. Comparison is done on the literal text of each token, so "2" and "2.0" are treated as different members.
The calculator builds five derived sets. The union A ∪ B contains every element that appears in either set. The intersection A ∩ B contains the elements common to both. The difference A ∖ B keeps the elements of A that are not in B, while B ∖ A does the reverse. The symmetric difference A △ B combines the two one-sided differences — the elements that belong to exactly one of the sets. The operation selector simply chooses which of these to show as the headline result.
The cardinalities follow directly: |A|, |B|, |A ∪ B|, |A ∩ B| and |A △ B| are the element counts of the respective sets. By inclusion–exclusion they satisfy |A ∪ B| = |A| + |B| − |A ∩ B|. The donut chart partitions the union into three disjoint regions — elements only in A, elements in both, and elements only in B — which is the standard two-circle Venn breakdown. If both boxes are empty the result is undefined; an empty individual set is allowed and yields the empty set { }.
Frequently asked questions
Use commas, spaces or new lines — any mix works. For example "1, 2 3\n4" is read as four elements.
No. A set holds unique members, so repeated tokens within a box are merged into a single element before any operation runs.
A △ B is the set of elements in exactly one of the two sets: (A ∖ B) ∪ (B ∖ A). Its size equals |A ∪ B| − |A ∩ B|.
Also known as
TG we-Calculate Editorial Team. (2026). Set Operations Calculator (Union & Intersection) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/set-operations-calculator
TG we-Calculate Editorial Team. "Set Operations Calculator (Union & Intersection)." TG we-Calculate. 2026. https://we-calculate.com/calculator/set-operations-calculator.
TG we-Calculate Editorial Team, "Set Operations Calculator (Union & Intersection)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/set-operations-calculator
@misc{wecalculate_set_operations_calculator, title = {Set Operations Calculator (Union & Intersection)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/set-operations-calculator}}, year = {2026}, note = {TG we-Calculate} }
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