An interval that represents a set of members by including both lower and higher values is called a closed interval.

According to the Set Theory, the members (or elements) are collected to represent their collection as a set. Actually, all elements lie between two members. Hence, all members can be expressed as an interval of two members. In this case, a set should be expressed by including the lower and higher value members and it is called a closed interval.

$x \ge a$ and $x \le b$

The two inequalities tell the following two factors in mathematical form.

- The value of $x$ is greater than or equal to $a$.
- The value of $x$ is less than or equal to $b$.

For our convenience, the two mathematical statements can also be written as follows.

$a \le x \le b$

This single mathematical inequality expresses that the value of $x$ lies between $a$ and $b$, and also equals to $a$ and $b$. Hence, this mathematical inequality is written as a closed interval between $a$ and $b$.

In mathematics, a closed interval is represented in two different ways.

A closed interval is represented in graphical system by considering the following two factors.

- An interval is graphically represented by a number line.
- The endpoints of the number-line are represented by the filled circles (or darkened circles) for saying that the lower and higher values are also considered.

A closed interval is represented in mathematical form by considering the following two factors.

- The lower and higher quantities are written in a row and they are separated by a comma $(,)$ and some space between them.
- A pair of square brackets are written before and after the lower and higher value elements to tell that their values are also included.

Therefore, a closed interval between $a$ and $b$ is written as $x \,∈\, \big[a, b\big]$

As per the set builder notation, a closed interval between $a$ and $b$ is written in the following forms.

$(1).\,\,\,$ $\Big\{x \,\,|\,\, x \,∈\, R \,\, and \,\, a \, \le \, x \, \le \, b\Big\}$

$(2).\,\,\,$ $\Big\{x \,:\, x \,∈\, R \,\, and \,\, a \, \le \, x \, \le \, b\Big\}$

Evaluate $f(x)$ if $f(x) = x+1$ where $x \,∈\, \big[2, 5\big]$

Let’s understand the concept of closed interval from this simple example. In this problem, the value of the function has to evaluate for every value of $x$ but the value of $x$ should be from $2$ to $5$. So, find the value of function $f(x)$ by substituting the value of $x$ from $2$ to $5$.

$f(2) \,=\, 2+1 = 3$

$f(3) \,=\, 3+1 = 4$

$f(4) \,=\, 4+1 = 5$

$f(5) \,=\, 5+1 = 6$

Latest Math Topics

Latest Math Problems

A best free mathematics education website for students, teachers and researchers.

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Learn how to solve the maths problems in different methods with understandable steps.

Copyright © 2012 - 2021 Math Doubts, All Rights Reserved