SET BUILDER NOTATION

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”

Examples :

(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}

Solved Problems

Problem 1 :

Describe the following sets in set builder form.

A  =  {1, 2, 3, 4, 5, 6}

Solution :

The given set contains set of natural numbers. But it ends up with 6. So, we may represent the given set in set builder form as follows.

A  =  {x : x is a natural number ≤ 6}

Problem 2 :

Write the following set in roster form

B  =  {1, 1/2, 1/3, 1/4, 1/5,................}

Solution :

The elements of the above set is in the form 1/n. The denominator is set of natural numbers.So, we may represent the given set in set builder form as follows.

B = {x : x = 1/n, n ∈ N}

Problem 3 :

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.

n = 1

x = 2n

x = 21

x = 2

n = 2

x = 2n

x = 22

x = 4

n = 3

x = 2n

x = 23

x = 8

n = 4

x = 2n

x = 24

x = 16

n = 5

x = 2n

x = 25

x = 32

X  =  {2, 4, 8, 16, 32}

Problem 4 :

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 }

Problem 5 :

Represent the following sets in set-builder form

A  =  {January, February, March, ............, December}

Solution :

A  =  {January, February, March, ............, December}

The set A contains all the months of a year.

Hence in set builder form, we write

X  =  {x : x is a months of a year}

Problem 6 :

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}

Problem 7 :

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}

Problem 8 :

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}

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