ROSTER FORM AND SET BUILDER FORM

Roster Form :

Listing the elements of a set inside a pair of braces { } is called the roster form.

(i) Let A be the set of even natural numbers less than 11.

In roster form we write A = {2, 4, 6, 8, 10}

(ii) A = {x : x is an integer and- 1≤  x < 5}

In roster form we write A = {-1, 0,1, 2, 3, 4}

Let us look into some examples in roster form.

Set Builder Form :

Set-builder notation is a notation for describing a set by indicating the properties that its members must satisfy.

Reading Notation :

‘|’or ‘:’ such that

A = { x : x is a letter in the word dictionary }

We read it as

“A is the set of all x such that x is a letter in the word dictionary”

For example,

(i) N = "x : x is a natural number,

(ii) P = "x : x is a prime number less than 100,

(iii) A = "x : x is a letter in the English alphabet,

Here we are going to see examples on roster form and set builder form.

Roster Form Examples

Example 1 :

List the elements of the following set in Roster form:

The set of all positive integers which are multiples of 7

Solution : 

The set of all positive integers which are multiples of 7 in roster form is

{7, 14, 21, 28,...........}

Example 2 :

List the elements of the following set in Roster form:

The set of all prime numbers less than 20.

Solution : 

The set of all prime numbers less than 20 in roster form is

{2, 3, 5, 7, 11, 13, 17, 19}

Example 3 :

Write the set A = { x : x is a natural number ≤ 8} in roster form.

Solution :

A = { x : x is a natural number ≤ 8}.

So, the set contains the elements 1, 2, 3, 4, 5, 6, 7, 8.

Hence in roster form A = {1, 2, 3, 4, 5, 6, 7, 8}

Example 4 :

Write the following set in roster form

A = {x : x ∈ N, 2 < x ≤ 10}

Solution :

A = {x : x ∈ N, 2 < x ≤ 10}

Set A will contain elements greater than 2 and less than or equal to 10.

A = { 3, 4, 5, 6, 7, 8, 9, 10}

Example 5 :

Write the following set in roster form

X =  {x : x = 2ⁿ, n ∈ N and n  ≤  5}

Solution :

X =  {x : x = 2ⁿ, n ∈ N and n  ≤  5}

To find the elements in the given set, we need to apply the values 1, 2, 3, 4 ,5 respectively instead of n.

X = { 2, 4, 8, 16, 32}

Set Builder Form Examples

Example 1 :

Represent the following sets in set-builder form

X = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}

Solution :

X = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}

The set X contains all the days of a week.

Hence in set builder form, we write

X = { x : x is a day in a week }

Example 2 :

Represent the following sets in set-builder form

A = {1, 1/2, 1/3, 1/4, ...............}

Solution :

A = {1, 1/2, 1/3, 1/4, ...............}

The denominators of the elements are 1, 2, 3, 4, ......

The set-builder form is A = { x : x ,1/n, n ∈  N }

Example 3 :

Write the following sets in Set-Builder form

 The set of all positive even numbers

Solution :

A = The set of all positive even numbers

The set-builder form is

A = { x : x is a positive even number}

Example 4 :

Write the following sets in Set-Builder form

The set of all whole numbers less than 20

Solution :

A = The set of all whole numbers less than 20

The set-builder form is

A = {x : x is a whole number and x < 20}

Example 5 :

Write the following sets in Set-Builder form

The set of all positive integers which are multiples of 3

Solution :

A = The set of all positive integers which are multiples of 3

The set-builder form is

A = {x : x is a positive integer and multiple of 3}

Example 6 :

Set A = {0, 7, 26, 63} in set-builder form is:

(a) {x: x ∈ N, x = n³ – 1 and n ≤ 7}

(b) {x: x ∈ N, x = n² – 1 and n ≤ 5}

(c) {x: x ∈ N, x = n² – 1 and n < 4}

(d) {x: x ∈ N, x = n³ – 1 and n ≤ 4}

Solution :

By observing the elements of set A, it is clear that these values are 1 less than perfect cube.

x = n³ – 1

So, option d is correct.

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