**Intervals as subsets of ℝ**

Let *a*, *b*∈ℝ and *a*<*b*. Then

(a) An open interval denoted by (*a*,*b*) is the set of real numbers {*x*:*a*<*x*<*b*}

(b) A closed interval denoted by [*a*,*b*] is the set of real numbers {*x*:*a*≤*x*≤*b*}

(c) Intervals closed at one end and open at the other are given by

[*a*,*b*)={*x*:*a*≤*x*<*b*}

(*a*,*b*]={*x*:*a*<*x*≤*b*}

⛲ Example 1. Write the following as intervals:

(i) {*x*:*x*∈ℝ, -4<*x*≤6}

(ii) {*x*:*x*∈ℝ, -12<*x*<-10}
(iii) {*x*:*x*∈ℝ, 0≤*x*<7}
(iv) {*x*:*x*∈ℝ, 3≤*x*≤4}

🔑

(i) {*x*:*x*∈ℝ, -4<*x*≤6}=(-4,6]

(ii) {*x*:*x*∈ℝ, -12<*x*<-10}=(-12,-10)
(iii) {*x*:*x*∈ℝ, 0≤*x*<7}=[0,7)
(iv) {*x*:*x*∈ℝ, 3≤*x*≤4}=[3,4]

⛲ Ex2. Write the following subset of ℝ as interval. Also find the length of interval and represent on number line.

*x*:

*x*∈ℝ, -12≤

*x*≤-10}

🌟 if inequalities are of the form ≥ or ≤, then use the symbol of closed interval and then find the length of the interval, which is equal to the difference of its extreme values.

✍ Solution:

*x*:

*x*∈ℝ, -12≤

*x*≤-10}=[-12,-10]

and length of interval =-10-(-12)=2.

On the real line set [-12,-10] may be graphed as shown in figure given below

The dark portion on the number line is the required set.

⛲ Ex3. (a) Write the following as intervals

(i) {*x*:*x*∈ℝ, -5<*x*≤6} (ii) {*x*:*x*∈ℝ,-11<*x*<-9}

(b) Write thefollowing as intervals and also represent on the number line.

(i) {*x*:*x*∈ℝ, 2≤*x*<8} (ii) {*x*:*x*∈ℝ,5≤*x*≤6}

🌟 If an inequality is of the form ≤ or ≥, then We use the symbol of closed interval, otherwise we use the symbol of open interval.

✍ Solution:

(a) (i) {*x*:*x*∈ℝ, -5<*x*≤6} is the set that does not contain -5 but contains 6. So, it can be written as an interval whose first end point is open and last end point is closed i.e., (-5,6].

(ii) {*x*:*x*∈ℝ,-11<*x*<-9} is the set that neither contains -11 nor -9, so it can be represented as open interval i.e., (-11,-9).

(b) (i) {*x*:*x*∈ℝ, 2≤*x*<8} is the set that contains 2 but not contain 8.50, it can be represented as an interval whose first end point is closed and the other end point is open i.e., [2,8). On the real line [2,8) may be graphed as shown in figure given below

The dark portion on the number line is the required set.

(ii) {*x*:*x*∈ℝ, 5≤*x*≤6} is the set which contains 5 and 6 both. So, it is equivalent to a closed interval i.e., [5,6]. On the real line [5,6] may be graphed as shown in the figure given below

The dark portion on the number line is the required set.