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An alternative interpretation of the Jaccard distance is as the ratio of the size of the symmetric difference = () to the union. Jaccard distance is commonly used to calculate an n × n matrix for clustering and multidimensional scaling of n sample sets.
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. A finite union is the union of a finite number of sets; the phrase does not imply that the union set is a finite set ...
In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events. This inequality provides an upper bound on the probability of occurrence of at least one ...
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 power set of a set S, together with the operations of union, intersection and complement, is a Σ-algebra over S and can be viewed as the prototypical example of a Boolean algebra. In fact, one can show that any finite Boolean algebra is isomorphic to the Boolean algebra of the power set of a finite set.
The total length of any covering interval set may overestimate the measure of , because is a subset of the union of the intervals, and so the intervals may include points which are not in . The Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets.
This Hasse diagram depicts a partially ordered set with four elements: a, b, the maximal element a b equal to the join of a and b, and the minimal element a b equal to the meet of a and b.
More generally, one can consider the fundamental groupoid on a set A of base points, chosen according to the geometry of the situation; for example, in the case of the circle, which can be represented as the union of two connected open sets whose intersection has two components, one can choose one base point in each component.