Open and Closed Interval

Open and Closed Interval are all numbers between given two numbers. The end numbers may be included or excluded. Collection of discreate numbers is called as sets, where as interval is continuous collection of all numbers between two numbers is a continuous system.

Mainly we consider ( Study ) intervals while evaluating inequalities. like

$\lt$ less than.
$\leq$ less than or equal to.
$\gt$ greater than.
$\geq $ greater than or equal to

End point may or may not be included. Based on end points are include or exclude the interval are classified as

Open Interval
Closed Interval

Open Interval

Open interval is in which end points are not included. The numbers are written as order pair between plain bracket i.e open interval a to b open is written as (a, b). Every real number from a to b is includes except a and b itself.

We get open interval when we use inequality like

$\lt $ less than
$\gt $ greater than

For example

All values of x greater than -3 and less than 5 are between -3 and 5 but -3 and 5 are not included $x\in\,(-3,5)$.
Values of x that satisfy the inequation $x-\sqrt{2} \gt 1$ is written as $x\in (1+\sqrt{2} ,\infty)$.
All values of x that satisfy inequation $x\lt -\sqrt{5}$ will be written as $x\in (-\infty,\,-\sqrt{5})$.

Closed Interval

Closed interval is in which end points are included. The numbers are written as pair between square bracket i.e Closed from a to b closed is written as [a, b]. Every real number from a to b including a and b are included.

Closed interval is used while writing inequalities

$\leq $ less than or equal to
$\geq $ greater than or equal to.

For example

All values of x greater than or equal to -3 and less than or equal to 5. In this both -3 and 5 are included, is written as $x\in\,[-3,5]$.
Values of x that satisfy the inequation $x-\sqrt{2} \geq 1$ is written as $x\in [1+\sqrt{2} ,\infty)$.
All values of x that satisfy inequation $x\leq -\sqrt{5}$ will be written as $x\in (-\infty,\,-\sqrt{5}]$

Remember

Only those vales be included in closed interval which satisfy the inequality and which are well defined number ration or irrational. e.g -3, 4, 5 ,$\sqrt{3}$ ….
The vales which satisfy the inequality but are not known to us always open. Like 2<x<7 is written as $x\in (2,7)$ since we do not know exactly what is a number just greater than 2 and just less than 7.
Infinity and negative infinity are always open interval. Because exact value of infinity or negative infinity is not known.

Operations with Open and Closed Interval

The different operations as discussed in sets are also performed on intervals.

Union Of Intervals

The values of x that satisfy either of inequality is union of interval.

e.g The vales of x that satisfy either $-1\leq x\lt 3$ OR $0\lt x \lt 7$ will be Union of two intervals i.e $[-1,3)\cup (0,7)$ will be [-1,7). Hence $x\in [-1,7)$

Intersection Of Intervals

The values of x that satisfy both the inequalities simultaneously is intersection of both intervals intervals.

e.g The vales of x that satisfy both inequations $-1\leq x\lt 3$ And $0 \lt x \lt 7$ will be intersection of two intervals i.e $[-1,3)\cap (0,7)$ will be (0,3). Hence $x\in (0,3)$.