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Interval

A set I of real numbers is an interval if it has the following special property:
Whenever a and c are in I and a < b < c, then c is in I.
There are several types of intervals. Suppose first that a and b are real numbers. Here are four different kinds of intervals of finite length.
open interval: (a, b) = {x in R : a < x < b}
closed interval: [a, b] = {x in R : a <= x <= b}
half-open interval: (a, b] = {x in R : a < x <= b}
other half-open interval: [a, b) = {x in R : a <= x < b}
There are five kinds of intervals of infinite length, sometimes called rays.
(a, +infinity) = {x in R: a < x}
[a, +infinity) = {x in R: a <= x}
(-infinity, a) = {x in R: x < a}
(-infinity, a] = {x in R: x <= a}
(-infinity, infinity) = R
Make particular note of the fact that infinity is not a real number, and neither is -infinity. The word infinity is simply used as a convenient label for a useful concept.

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