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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 ...
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.
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 ...
4 members of a sequence of nested intervals. In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of intervals on the real number line with natural numbers =,,, … as an index. In order for a sequence of intervals to be considered nested intervals, two conditions have to be met:
3. An ideal is called κ-complete if it is closed under the union of less than κ elements 4. A measure is called κ-complete if the union of less than κ measure 0 sets has measure 0 5. A linear order is called complete if every nonempty bounded subset has a least upper bound Con Con(T) for a theory T means T is consistent condensation lemma
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.
The notation is also used to denote the characteristic function in convex analysis, which is defined as if using the reciprocal of the standard definition of the indicator function. A related concept in statistics is that of a dummy variable .