Is Every Interval a Set

An interval is a set of (real) numbers with the property that for every pair of numbers y and z that it contains, it contains also every number between y and z.

What makes a set an interval?

A real interval is a set of real numbers with the property that any number that lies between two numbers included in the set is also included in the set. The interval of numbers between a and b , including a and b , is denoted [a,b] . … An interval is unbounded if both endpoints are not real numbers.

Can an interval be a subset of a set?

Intervals: An interval is a subset I of R∗ satisfying the following requirement: Whenever x < y < z and x,z ∈ I then y ∈ I. Obvious examples are the sets we have been taught to consider intervals: (a,b) = {x : a < x < b}, [a,b] = {x : a ≤ x ≤ b}, (a,b] = {x : a < x ≤ b}, [a,b) = {x : a ≤ x < b}.