 Representation of a Set – Statement Form, Set Builder Form, Roster Form

Interested students can see the process of representation of sets on this page. We have three different ways to represent a set. Get the solved examples on sets in the following sections. Check out the Representation of a Set in set-builder form, roster form, and statement form from the below sections of this page. Also, refer to solved examples on the representation of a set using different notations explained clearly.

What is Meant by Representation of a Set?

Sets are the collection of well-defined objects. The numbers, alphabets and others enclosed between the curly braces of a set are called the elements. The elements are separated by a comma symbol. Usually, sets are denoted by capital letters i.e A, B, C and so on. We have three ways for representing a set, they are

1. Descriptive Form

2. Set Builder Form

3. Roster Form

Descriptive Form

It is a way of representing a set in the verbal statement. It gives a description of elements in the set. The description must allow a concise determination of which elements belong to the set and which elements do not.

Examples:

• The set of natural numbers less than 25.
• The set of vowels in the alphabets.
• The set of all letters in English alphabets.
• The set of prime numbers less than 50.
• The set of even numbers between 20 and 40.

Roster Form or Tabular Form

Roster form means listing all the elements of a set inside a pair of curly braces {}.

Examples:

The natural numbers less than 15 are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14.

Let N be the set of natural numbers less than 15.

N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}

The prime numbers lesser than 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Let P be the set of prime numbers below 50.

P = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47}

Let M be the set of all months in a year.

Therefore, M = {January, February, March, April, May, June, July, August, September, October November, December}

Set-Builder Form or Rule Form

In the set-builder form, the statements are written inside a pair of braces. In this case, all the set elements must have a single property to become the set member. Here, the set elements are described by a symbol ‘x’ or any other variable followed by a colon “:” or slash “|”. After writing the symbol, you need to write a statement including the variable. In this, colon or slash stands for such that and braces stands for ‘set of all’.

Examples:

(i) Let P is the set of natural numbers between 15 and 25.

The set builder form is

P = { x : x is a natural number between 15 and 25 } or

P = { x | x is a natural number between 15 and 25 }

You can read this as P is a set of elements x such that x is a natural number between 15 and 25.

(ii) Let A denote the set of prime numbers between 5 and 50. It can be written in the set builder form as

A = { x | x is a prime number, 5 < x < 50 }

or A = { x : x ∈ P, 5 < x < 50 and P is an prime number }

(iii) The set B of all even natural numbers can be written as

B = {x : x is a natural number and x = 2n for n ∈ W}

Example Questions on Set Representation Using 3 Methods

Question 1:

The set of days of a week.

Solution:

Given that,

Set of days of a week.

The statement form is Set of seven days in a week.

The days in a week are Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday

The roster form is W ={Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}

The set-builder notation is W = { x : x is a day of the week }

Question 2:

The set of whole numbers lying between 5 and 25.

Solution:

Given that,

The set of whole numbers lying between 5 and 25.

The description notation is Set of whole numbers between 5 and 25.

The whole numbers lying between 5 and 25 are 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, and 24.

The set-builder form is A = {x | x is a whole number, 5 < x <24}

The roster form is A = {6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}

Question 3:

The set of all numbers lesser than 16 and greater than 8.

Solution:

Given that,

The set of all numbers lesser than 16 and greater than 8.

The numbers greater than 8 and less than 16 are 9, 10, 11, 12, 13, 14, 15

The roster form is N = {9, 10, 11, 12, 13, 14, 15}

The statement form is the set of numbers between 8 and 16.

The set builder form is N = { x : x ∈ A, 8 < x < 16, A is a natural number}

FAQs on Representation of a Set

1. What are the ways for representing a set?

The 3 various ways of set representation are statement form or description form, set-builder form or rule form, roster form or tabular form.

2. What is the formula to use rule form?

The rule form formula is { x : property}. Here property defines the elements of a set.

3. What is the best way to represent sets?

According to me, the best and most used way of writing a set is roster form. The advantage of using the roster form is we can just list the set elements between the curly braces and each element is separated by a comma.

4. What are the two methods of writing sets?

The two main methods of representation of a set are using a Venn diagram or listing the elements (roster form). Venn diagram is the pictorial representation and roster form is the mathematical representation.

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