Mathematics & Statistics 1 – open and closed intervals

An interval is a set of real no. which is lying between any 2 numbers of the set. The intervals can be continuous or discontinuous in nature.

As shown in the image, an interval is the set of numbers lying between the point ‘x’ and ‘y’. The two numbers ‘x’ and ‘y’ are known as the interval’s endpoints or bounding points. The interval is denoted using a parenthesis ‘()’ or a square bracket ‘[]’.

Types of Intervals

The intervals are classified into three categories based on the inclusion or exclusion of the endpoints. The type of intervals are as follows – 

1. Open interval

2. Closed interval

3. Semi-closed or semi-open interval

Open Interval

In the open interval, the endpoints are not included in the set. The open interval is denoted using parenthesis ‘()’. 

The given image is a pictorial representation of the open interval. The endpoints ‘x’ and ‘y’ are not to be included in the set. The interval should be denoted as (x,y). The algebraic expression for the open interval is x<A<y.

Problems based on open interval: 

1. Find the algebraic expression of the interval (1,6) – 

The interval (1,6) includes all real numbers between 1 and 6, but not 1 and 6. It can be denoted as 1<x<6

2. Find the elements of the algebraic expression 12<x<15

In the given expression, the endpoints 12 and 15 are not included in the interval. Hence it is an example of an open interval

12<x<15 = (12,15)

3. Write the algebraic expression 5<x<9 in interval notation

In the given expression, the endpoints 5 and 9 are not included in the data set. Hence it is an open interval. Therefore the interval notation of the algebraic expression 5<x<9 is (5,9).

Closed Interval

In closed intervals, the endpoints are included in the set. The open interval is denoted using parenthesis ‘[]’. 

The given image is a pictorial representation of the closed interval. The endpoints ‘x’ and ‘y’ are included in the set. The interval should be denoted as [x,y]. The algebraic expression for the closed interval is x≤A≤y. 

Problems based on closed interval: 

1. Find the algebraic expression of the interval [1,6] – 

The interval [1,6] includes all the real numbers between 1 and 6, including 1 and 6. It can be denoted as 1≤x≤6

2. Find the elements of the algebraic expression 12≤x≤15

In the given expression, the endpoints 12 and 15 are to be included in the interval. Hence it is an example of closed interval

12≤x≤15 = [12, 15]

3. Write the algebraic expression 5≤x≤9 in interval notation

In the given expression, the endpoints 5 and 9 are included in the data set. Hence it is a closed interval. Therefore the interval notation of the algebraic expression 5≤x≤9 is [5,9].

Semi-Closed or Semi-Open Interval

Semi-closed intervals are also termed semi-open intervals in which one endpoint is included in the interval and another endpoint is not included in the interval. It can be denoted with one parenthesis and one square bracket – (x,y] or [x,y). 

The semi-closed interval is further classified into two categories – 

1. ‘left-closed, right-open’ interval

2. ‘right-closed, left-open’ interval

The given image is a pictorial representation of a semi-closed interval in which the endpoint ‘x’ is included in the interval, whereas the endpoint ‘y’ is not included in the interval.  The interval should be denoted as [x,y). It can also be termed as ‘left-closed, right-open’ interval. The algebraic expression for the semi-closed interval is x≤A<y. 

The given image is a representation of a semi-closed interval in which the endpoint ‘x’ is not included in the interval, whereas the endpoint ‘y’ is included in the interval.  The interval should be denoted as (x,y]. It can also be termed as ‘left-open, right-closed’ interval. The algebraic expression for the semi-closed interval is x<A≤y. 

Problems based on a semi-closed or semi-open interval: 

1. Find the algebraic expression of the interval (1,6] – 

The interval (1,6] includes all the real numbers between 1 and 6, excluding 1 and including 6. It can be denoted as 1<x≤6

2. Find the elements of the algebraic expression 12<x≤15

In the given expression, the endpoint 12 is not to be included in the interval whereas the endpoint 15 is to be included in the interval. Hence it is an example of a semi-closed interval

12<x≤15 = (12,15]

3. Write the algebraic expression 5<x≤9 in interval notation

In the given expression, the endpoint 5 is not included in the data set, whereas the endpoint 9 is included in the data set. Hence it is a semi-closed interval. Therefore the interval notation of the algebraic expression 5<x≤9 is (5,9].

4. Write the algebraic expression 16≤x<19 in interval notation

In the given expression, the endpoint 16 is included in the data set, whereas the endpoint 19 is not included in the data set. Hence it is a semi-closed interval. Therefore the interval notation of the algebraic expression 16≤x<19 is [16,19).

Conclusion 

The interval is defined as the numbers lying between two end points/boundary points. Based on the inclusion/exclusion of endpoints, the intervals are classified as – closed, open and semi-closed/semi-open intervals.