What is the formula for two sets?

|A ∪ B| = |A| + |B| − |A∩B| — you subtract the intersection once because simply adding |A| + |B| counts every element in both sets twice.

What is the formula for three sets?

|A ∪ B ∪ C| = |A| + |B| + |C| − |A∩B| − |A∩C| − |B∩C| + |A∩B∩C| — the triple intersection is added back because it gets subtracted three times by the pairwise terms but was only counted three times in the individual sizes, requiring one addition to correct the balance.

Why do I need the triple intersection for 3 sets but not for 2?

With only two sets there's no 'triple overlap' to correct for — the single pairwise subtraction fully accounts for double-counting. With three sets, the pairwise subtractions overcorrect the elements in all three sets, so the triple intersection term restores the balance.

What if I don't know the exact elements, only counts?

That's exactly what this calculator is for — it works purely from the sizes of each set and each intersection, without needing the underlying element lists, unlike a full set-listing tool.

Can intersection sizes be larger than the sets themselves?

No — this calculator validates that each intersection size doesn't exceed the smaller of the sets (or pairwise intersections) it's derived from, and reports an error if the numbers are inconsistent.

Does this support more than 3 sets?

Not currently — this calculator is scoped to 2 or 3 sets, which covers the overwhelming majority of practical inclusion-exclusion problems. Beyond 3 sets, the number of intersection terms grows quickly (2ⁿ − 1 terms for n sets).

Where is inclusion-exclusion used in real life?

It's used in probability (computing the probability of at least one of several events), database query counting, and combinatorics problems like counting numbers divisible by at least one of several given divisors.

Inclusion-Exclusion Calculator — Find the Size of a Union of Sets

Q: What is the inclusion-exclusion principle?

It's a counting technique for finding the size of a union of overlapping sets by alternately adding and subtracting the sizes of all possible intersections, correcting for elements counted multiple times.

Our Inclusion-Exclusion Calculator computes |A ∪ B| or |A ∪ B ∪ C| directly from the sizes of the sets and their pairwise (and triple, for 3 sets) intersections — no element lists required, just counts — applying the inclusion-exclusion principle exactly and showing the full formula substitution.

Quick Answer

Enter the sizes of your sets and their intersections below to instantly compute the size of their union via the inclusion-exclusion principle, for 2 or 3 sets.

|A|

|B|

|A∩B|

How to Use the Inclusion-Exclusion Calculator — Union of 2 or 3 Sets Online

1
Choose 2 sets or 3 sets.
2
Enter the size of each set and the size of each intersection (and the triple intersection, for 3 sets).
3
Click 'Calculate' to see the size of the union and the formula substitution.

Why Use Inclusion-Exclusion Calculator — Union of 2 or 3 Sets Online?

Simply adding up the sizes of overlapping sets double-counts every element that appears in more than one — the inclusion-exclusion principle corrects for that by alternately adding and subtracting the sizes of every possible intersection: add the individual sets, subtract the pairwise overlaps (which were counted twice), then add back the triple overlap (which got subtracted out entirely by the pairwise correction). This calculator works purely from counts rather than actual element lists, making it useful when you know set sizes and overlap sizes from survey data or other aggregate counts rather than the underlying elements themselves — distinct from the site's Set Theory Calculator, which operates on explicit lists of elements.

Frequently Asked Questions

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