Union, intersection, differences and symmetric difference of two sets of numbers or words, with the Jaccard index and a Venn diagram.
Cập nhật Ví dụ đã kiểm tra: 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.
Nguồn đối chiếu: Python set operators |, &, -, ^ on {1,2,3,4,5} and {4,5,6,7}; 2/7 = 0.285714
Nguồn đối chiếu: Python set operators on string sets, sorted()
Nguồn đối chiếu: Python: {1,3,5} & {2,4,6} == set()
Nguồn đối chiếu: Python: {Decimal('1'), Decimal('1.0'), Decimal('2')} has 2 elements; Decimal('2.00') == Decimal('2')
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.
Độ chính xác phụ thuộc vào dữ liệu nhập và giả định của phương pháp. Phép tính thập phân dùng 50 chữ số có nghĩa, nhưng ước lượng, phương pháp số và dữ liệu nguồn có thể kém chính xác hơn; làm tròn khi hiển thị không loại bỏ các giới hạn đó. Ví dụ có lời giải đã đối chiếu với nguồn độc lập: 6. Ví dụ, “{1, 2, 3, 4, 5} and {4, 5, 6, 7}” được kiểm tra bằng Python set operators |, &, -, ^ on {1,2,3,4,5} and {4,5,6,7}; 2/7 = 0.285714.
Wolfram MathWorld — Union, Intersection, Set Difference, Symmetric Difference; Python documentation — set types and operations.
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