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.