Intervals

Bounded intervals

We will call interval the set of numbers included between two given limits.

If $a$ and $b$ are two real numbers such that $a\le b$, the interval of endpoints $a$ and $b$ is the segment $\overline{ab}$, or, also, the set of numbers included between $a$ and $b$.

If we consider that the endpoints $a$ and $b$ belong to the interval, we will say that it is a closed interval and will denote it as $[a,b]$.

If $x$ is a real number that belongs to $[a,b]$, the point that represents $x$ on the line is on the right of $a$ and on the left of $b$; this means that $a<x<b$, and since $a$ and $b$ belong to the interval as well, it is possible that $x=a$ or $x=b$, so a real number $x$ belongs to the closed interval $[a,b]$ if $a\le x\le b$. We will write this algebraic definition in the following way: $$[a,b]=\{x\in \mathbb{R}|a\le x\le b\}$$

If the endpoints do not belong to the interval, we call it an open interval and we will denote it as $(a,b)$. If $x$ is a real number that belongs to $(a,b)$, it is necessary that $a<x<b$, and we will write it in algebraic language as: $$(a,b)=\{x\in \mathbb{R}|a<x<b\}$$

If only one of the endpoints belongs to the interval we say that it is a semiopen interval and we will denote it as $(a,b]$ or $[a,b)$, depending on which endpoint belongs to the interval:

$$(a,b]=\{x\in \mathbb{R}|a<x\le b\}$$ $$[a,b)=\{x\in \mathbb{R}|a\le x<b\}$$

In any kind of interval, $a$ is the lower endpoint , and $b$ the upper endpoint. And $|b-a|$ is the length of the interval.

The point $C$ at the same distance from $a$ to $b$, we will call center of the interval. We will call the distance between the center of the interval and the endpoints the radius .

The center of an interval of endpoints $a$ and $b$ is the point ${\displaystyle \frac{a+b}{2}}$; in fact:

$$d(a,{\displaystyle \frac{a+b}{2}})=|{\displaystyle \frac{a+b}{2}}-a|=|{\displaystyle \frac{a+b-2a}{2}}|={\displaystyle \frac{b-a}{2}}$$

$$d({\displaystyle \frac{a+b}{2}},b)=|b-{\displaystyle \frac{a+b}{2}}|=|{\displaystyle \frac{2b-a-b}{2}}|={\displaystyle \frac{b-a}{2}}$$

On the other hand, the points of an interval of endpoints $a$ and $b$ can be defined in terms of the distance to the center of the interval.

If $x\in [a,b]$, the distance of $x$ to the center is less than or equal to the radius of the interval, and as $d(x,C)=|C-x|$, we have: $$[a,b]=\{x\in \mathbb{R}||C-x|\le r\}$$ where $r$ represents the radius of the interval $(r=d(a,b))$, and, similarly, for open intervals: $$(a,b)=\{x\in \mathbb{R}||C-x|<r\}$$

To determine the endpoints of an interval given the center and the radius, we apply the properties of the absolute value:

$$|C-x|<r\Rightarrow |x-C|<r\Rightarrow$$ $$-r<x-C<r\Rightarrow -r+C<x<r+C$$

Therefore the endpoints of an interval of center $C$ and radius $r$ are $C-r$ and $C+r$.

The length of an interval is equal to the distance between its two endpoints: $$long([a,b])=d(a,b)$$ And as it depends on the endpoints, the length is the same whether the interval is open or closed: $$long((a,b))=long([a,b])=long((a,b])=long([a,b))$$

Let's observe that the length of an interval depends on the distance used when calculating it, so, continuing with the previous notation, if we use the p-adic distance to calculate the length of an interval, we will denote it as: $$lon{g}_p((a,b))=d_p(a,b)$$

Unbounded intervals

If we consider an interval that does not have lower endpoint or upper endpoint, we obtain a set of the kind: $$\{x\in \mathbb{R}|x\le b\},\text{or}\{x\in \mathbb{R}|a\le x\}$$

Graphically, these sets are represented as all those that are on the left of $b$, or on the right of $a$, respectively.

We call these sets unbounded intervals and to denote them we use the infinity symbol $\mathrm{\infty }$ as an endpoint. Although $\mathrm{\infty }$ is not a number, we will use $-\mathrm{\infty }$ to denote that it is less than any number and $+\mathrm{\infty }$ to denote that it is greater than any number, in such a way that a lower unbounded interval is denoted by:

$(-\mathrm{\infty },a)=\{x\in \mathbb{R}|x<a\}$

if it is opened, whereas if it is closed:

$(-\mathrm{\infty },a]=\{x\in \mathbb{R}|x\le a\}$

If the interval does not have upper endpoint, we call it an unbounded upper, and we write it as:

$(a,+\mathrm{\infty })=\{x\in \mathbb{R}|a<x\}$

if it is opened, and

$[a,+\mathrm{\infty })=\{x\in \mathbb{R}|a\le x\}$

if it is closed.