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One can take the union of several sets simultaneously. For example, the union of three sets A, B, and C contains all elements of A, all elements of B, and all elements of C, and nothing else. Thus, x is an element of A ∪ B ∪ C if and only if x is in at least one of A, B, and C.
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
Venn diagram showing the union of sets A and B as everything not in white. In combinatorics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as
The intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}. ... Intersection distributes over union and union distributes over ... MinHash – Data mining technique;
Python has built-in set and frozenset types since 2.4, and since Python 3.0 and 2.7, supports non-empty set literals using a curly-bracket syntax, e.g.: {x, y, z}; empty sets must be created using set(), because Python uses {} to represent the empty dictionary.
As with sets, and in contrast to tuples, the order in which elements are listed does not matter in discriminating multisets, so {a, a, b} and {a, b, a} denote the same multiset. To distinguish between sets and multisets, a notation that incorporates square brackets is sometimes used: the multiset {a, a, b} can be denoted by [a, a, b]. [2]
The axiom of replacement allows one to form many unions, such as the union of two sets. However, in its full generality, the axiom of union is independent from the rest of the ZFC-axioms: [ citation needed ] Replacement does not prove the existence of the union of a set of sets if the result contains an unbounded number of cardinalities.
More formally, given a universe and a family of subsets of , a set cover is a subfamily of sets whose union is . In the set cover decision problem , the input is a pair ( U , S ) {\displaystyle ({\mathcal {U}},{\mathcal {S}})} and an integer k {\displaystyle k} ; the question is whether there is a set cover of size k {\displaystyle k} or less.