Union, intersection, differences and symmetric difference of two sets of numbers or words, with the Jaccard index and a Venn diagram.
Actualizado Ejemplos verificados: 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.
Fuente de comprobación: Python set operators |, &, -, ^ on {1,2,3,4,5} and {4,5,6,7}; 2/7 = 0.285714
Fuente de comprobación: Python set operators on string sets, sorted()
Fuente de comprobación: Python: {1,3,5} & {2,4,6} == set()
Fuente de comprobación: 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.
La precisión depende de tus datos y de los supuestos del método. El cálculo decimal usa 50 cifras significativas, pero las estimaciones, los métodos numéricos y los datos de origen pueden ser menos precisos; el redondeo mostrado no elimina esos límites. Ejemplos resueltos comprobados con fuentes independientes: 6. Por ejemplo, «{1, 2, 3, 4, 5} and {4, 5, 6, 7}» se comprueba con 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.
Esta calculadora incluye 6 ejemplos resueltos con respuestas de fuentes independientes. Forman parte del conjunto de pruebas y también puedes ejecutarlos aquí.