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The union of two intervals is an interval if and only if they have a non-empty intersection or an open end-point of one interval is a closed end-point of the other, for example (,) [,] = (,]. If R {\displaystyle \mathbb {R} } is viewed as a metric space , its open balls are the open bounded intervals ( c + r , c − r ) , and its closed balls ...
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 ...
Universe set and complement notation The notation L ∁ = def X ∖ L . {\displaystyle L^{\complement }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~X\setminus L.} may be used if L {\displaystyle L} is a subset of some set X {\displaystyle X} that is understood (say from context, or because it is clearly stated what the superset X ...
Example: the blue circle represents the set of points (x, y) satisfying x 2 + y 2 = r 2.The red disk represents the set of points (x, y) satisfying x 2 + y 2 < r 2.The red set is an open set, the blue set is its boundary set, and the union of the red and blue sets is a closed set.
Occasionally, the notation is used for the disjoint union of a family of sets, or the notation + for the disjoint union of two sets. This notation is meant to be suggestive of the fact that the cardinality of the disjoint union is the sum of the cardinalities of the terms in the family.
Disjoint-set data structures [9] and partition refinement [10] are two techniques in computer science for efficiently maintaining partitions of a set subject to, respectively, union operations that merge two sets or refinement operations that split one set into two. A disjoint union may mean one of two things.
In 1977, Victor Klee considered the following problem: given a collection of n intervals in the real line, compute the length of their union.He then presented an algorithm to solve this problem with computational complexity (or "running time") () — see Big O notation for the meaning of this statement.
The complement of the union of these intervals is itself a union of a finite number of intervals, which we denote {J(ε) i} (for 1 ≤ i ≤ k − 1 and possibly for i = k, k + 1 as well). We now show that for every ε > 0 , there are upper and lower sums whose difference is less than ε , from which Riemann integrability follows.