Union, intersection, differences and symmetric difference of two sets of numbers or words, with the Jaccard index and a Venn diagram.
By AdminDrafted with AI writing toolsUpdated Checked examples: 6
A and B share 2 of the 7 distinct items (Jaccard index 0.2857); 3 are only in A and 2 only in B.
Jaccard index: 2 / 7 = 0.285714.
The calculator treats each list as a set, dropping repeats, and compares the two. The union A ∪ B holds everything in either set, the intersection A ∩ B what is in both, the differences A − B and B − A what is in one but not the other, and the symmetric difference A △ B what is in exactly one. The Jaccard index |A ∩ B| ÷ |A ∪ B| scores the overlap from 0 for disjoint sets to 1 for identical ones.
Comparing customer or email lists, finding shared tags, checking what one version of a list adds or removes, and discrete-mathematics homework are typical uses. The default sets {1, 2, 3, 4, 5} and {4, 5, 6, 7} share 2 of their 7 distinct items, so the Jaccard index is 2/7 ≈ 0.2857.
Items can be numbers or words, separated by commas, semicolons, new lines or spaces. Numbers are compared by value, so 1 and 1.0 are one item; words must match exactly, including capital letters. Results list numbers in order, then words alphabetically.
The union A ∪ B holds everything in either set, the intersection A ∩ B only what both share, and the difference A − B what A has that B lacks. For A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7}, A ∪ B = {1, 2, 3, 4, 5, 6, 7}, A ∩ B = {4, 5} and A − B = {1, 2, 3}.
Each element counts once, so 4 and 5 appear only once in the union even though both sets contain them.
These results use the sets above, A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7}.
| Operation | Meaning | Result for the default sets |
|---|---|---|
| A ∪ B | In either set or both | {1, 2, 3, 4, 5, 6, 7} |
| A ∩ B | In both sets | {4, 5} |
| A − B | In A but not B | {1, 2, 3} |
| B − A | In B but not A | {6, 7} |
| A △ B | In exactly one set | {1, 2, 3, 6, 7} |
Union, intersection and symmetric difference give the same answer in either order. The two differences do not: A − B = {1, 2, 3}, while B − A = {6, 7}. A − B is also written A \ B.
A △ B holds the elements that are in exactly one of the sets. It equals (A − B) ∪ (B − A), or the union with the intersection removed: {1, 2, 3, 6, 7} here.
The symmetric difference is the set version of exclusive or (XOR), and Python's ^ operator computes it (Python set documentation).
The size of a union is |A ∪ B| = |A| + |B| − |A ∩ B|, because elements in both sets would otherwise be counted twice. For the sets above, 5 + 4 − 2 = 7.
Word problems use the same rule. If 18 of 30 students play football, 12 play cricket and 5 play both, then 18 + 12 − 5 = 25 play at least one of the two, and 30 − 25 = 5 play neither.
For three sets the rule extends to |A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |A ∩ C| − |B ∩ C| + |A ∩ B ∩ C|.
The Jaccard index measures overlap as |A ∩ B| ÷ |A ∪ B|, from 0 for sets with nothing in common to 1 for identical sets. For the sets above it is 2/7 ≈ 0.2857, and the Jaccard distance is 1 − 2/7 = 5/7 ≈ 0.714.
It counts distinct elements only, so it ignores how often an item was listed and treats near-spellings as different items.
If A − B is empty, every element of A is also in B, so A is a subset of B (A ⊆ B).
A complement needs a universal set U: A′ = U − A. With U = {1, 2, …, 10} and A = {1, 2, 3, 4, 5}, A′ = {6, 7, 8, 9, 10}. B − A is the complement of A only when B is the whole universe.
The empty set ∅ has no elements: ∅ ∪ B = B and ∅ ∩ B = ∅. When both sets are empty, the Jaccard index would be 0 ÷ 0, so it is undefined.
For and, or and xor on true/false variables, use the truth table generator. For the probability version of inclusion–exclusion, P(A or B) = P(A) + P(B) − P(A and B), use the probability calculator.
Checked against: Set operators |, &, -, ^ on {1,2,3,4,5} and {4,5,6,7}; 2/7 = 0.285714
Checked against: Independent calculation
Checked against: {1,3,5} & {2,4,6} == set()
Checked against: Independent calculation
The union A ∪ B contains every element in A, in B or in both; the intersection A ∩ B contains only the elements in both. For A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7}, the union is {1, 2, 3, 4, 5, 6, 7} and the intersection is {4, 5}. The union is never smaller than either set, and the intersection never larger.
The elements that are in exactly one of the two sets: A △ B = (A − B) ∪ (B − A), which is the union with the intersection removed. For {1, 2, 3, 4, 5} and {4, 5, 6, 7} it is {1, 2, 3, 6, 7}. It behaves like XOR in logic, and Python's ^ operator computes it for sets.
Use the inclusion–exclusion principle: |A ∪ B| = |A| + |B| − |A ∩ B|, subtracting the intersection because its elements were counted twice. With |A| = 5, |B| = 4 and |A ∩ B| = 2, the union has 5 + 4 − 2 = 7 elements. For three sets it extends to |A| + |B| + |C| − |A ∩ B| − |A ∩ C| − |B ∩ C| + |A ∩ B ∩ C|.
A similarity score for two sets: the size of their intersection divided by the size of their union, from 0 for disjoint sets to 1 for identical ones. {1, 2, 3, 4, 5} and {4, 5, 6, 7} share 2 of 7 distinct items, a Jaccard index of 2/7 ≈ 0.2857. It is used to compare documents, tag sets and search results, and 1 minus the index is the Jaccard distance.
No. A − B, also written A \ B, keeps the elements of A that are not in B, while B − A keeps the elements of B that are not in A. With A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7}, A − B = {1, 2, 3} but B − A = {6, 7}. The two differences never share an element, and together they form the symmetric difference.
This calculator includes 6 worked examples with independently sourced answers. They run in the test suite, and you can run them here too.