# Union and intersection of intervals

Union and intersection of intervals

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A set is a collection of objects. Each object in the set is called an element of the set. If an element x is a member of the set S , we write. If an element x is not a member of the set S , we write. The empty set is the set that contains no elements. The union of two sets A and B , denoted is the set that consists of all elements that are in A or in B or in both A and B.

In mathematics, interval is a set of real numbers with the property that any number that lies between 2 numbers in the set is also included in the set. Real intervals play an important role in the theory of integration. Equations and inequalities are mathematical sentences formed by relating 2 expressions to each other. An equation or an inequality that contains at least 1 variable is called an open sentence. The 2 numbers are called the endpoints of the interval. To indicate that one of the endpoints is to be excluded from the set, the corresponding square bracket can be either replaced with a parenthesis, or reversed. Thus, in set builder notation:.

The relaxed intersection of m sets corresponds to the classical intersection between sets except that it is allowed to relax few sets in order to avoid an empty intersection. This notion can be used to solve Constraints Satisfaction Problems that are inconsistent by relaxing a small number of constraints. When a bounded-error approach is considered for parameter estimation , the relaxed intersection makes it possible to be robust with respect to some outliers. This definition is illustrated by Figure 1. Characterizing the q-relaxed intersection is a thus a set inversion problem.

When two inequalities are joined by the word and , the solution of the compound inequality occurs when both inequalities are true at the same time. It is the overlap, or intersection, of the solutions for each inequality. When the two inequalities are joined by the word or , the solution of the compound inequality occurs when either of the inequalities is true. The solution is the combination, or union, of the two individual solutions. In this section we will learn how to solve compound inequalities that are joined with the words AND and OR.

## Union, intersection and complementary of intervals

If you're seeing this message, it means we're having trouble loading external resources on our website. - All the Real Numbers that lie between those 2 values. In "Interval Notation" we just write the beginning and ending numbers of the interval, and use:.

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