What Is A Closed Interval

Closed interval

Math Doubts
Set Theory
Intervals

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

Introduction

According to the Set Theory, the members (or elements) are collected to represent their drove as a set up. Actually, all elements prevarication betwixt ii members. Hence, all members can be expressed as an interval of 2 members. In this instance, a set should be expressed by including the lower and higher value members and it is chosen a closed interval.

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 exist written as follows.

$a \le ten \le b$

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

Representation

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

Graphical Representation

A airtight interval is represented in graphical system by considering the following ii factors.

graphical representation of a closed interval

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

Mathematical Representation

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

mathematical representation of a closed interval

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 earlier and after the lower and college value elements to tell that their values are likewise included.

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

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

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

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

Case

Evaluate $f(x)$ if $f(10) = ten+1$ where $x \,∈\, \large[2, five\big]$

Let's empathise the concept of closed interval from this simple example. In this problem, the value of the part has to evaluate for every value of $x$ but the value of $10$ should be from $ii$ to $v$. So, find the value of office $f(10)$ by substituting the value of $x$ from $two$ to $five$.

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

$f(3) \,=\, three+ane = 4$

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

$f(five) \,=\, v+i = 6$