# Introduction to intervals

Look at the following sets of numbers: $$A=\{x\in\mathbb{R} \ | \ 2 < x < 5 \}$$$$$B=\{x\in\mathbb{R} \ | \ 2 \leq x \leq 5 \}$$$ $$C=\{x\in\mathbb{R} \ | \ 2 < x \leq 5 \}$$$$$D=\{x\in\mathbb{R} \ | \ 2 \leq x < 5 \}$$$

Note that the four sets contain only the points between $$2$$ and $$5$$ with the possible exceptions of $$2$$ and/or $$5$$. These sets are called intervals and the numbers $$2$$ and $$5$$ are the endpoints of each interval.

Moreover, $$A$$ is an open interval because it does not contain the endpoints; $$B$$ is a closed interval, it contains both endpoints, and sets $$C$$ and $$D$$ are neither open nor closed because they contain one of the two endpoints.

As intervals appear very often in mathematics, it is common to use a shorthand notation to describe intervals. For example, the previous intervals are denoted as:

$$A=(2,5)=]2,5[$$$$$B=[2,5]$$$ $$C=(2,5]=]2,5]$$$$$D=[2,5)=[2,5[$$$

## Properties of the intervals

Let $$\mathbb{R}$$ be the family of all intervals of the real line. Included in $$\mathbb{R}$$ are: the empty set $$\emptyset$$ and the points $$a = [a,a]$$. Intervals, then, have the folllowing properties:

1. The intersection of two intervals is an interval; that is, $$A,B \in \mathbb{R}\Rightarrow A\cap B\in\mathbb{R}$$.

2. The union of two no disjoint intervals is an interval; that is, $$A,B \in \mathbb{R}$$ and $$A\cap B\neq\emptyset \Rightarrow A\cup B\in\mathbb{R}$$.

3. The difference of two non comparable intervals is an interval; this is $$A,B \in \mathbb{R}$$ and $$A, B$$ not comparables $$\Rightarrow A-B\in\mathbb{R}$$.

## Infinite intervals

The sets of the form $$A=\{x \ | \ x > 1 \}$$$$$B=\{x \ | \ x \leq 0 \}$$$ $$C=\{x \ | \ x \in\mathbb{R}\}$$$are called infinite intervals and they are also denoted as $$A=(1,\infty)$$$ $$B=(-\infty,0)$$$$$C=(-\infty,\infty)$$$

## Bounded and unbounded sets

Let $$A$$ be a set of numbers; it is said that $$A$$ is a bounded set if $$A$$ is a subset of a finite interval. An equivalent definition is "Set $$A$$ is bounded if there is a positive number $$M$$, such that $$|x|\leq M, \ \forall x\in A$$". A set is said to be unbounded if it is not bounded.