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.

🌈 How to Write Intervals in set-builder form?