# SET BUILDER NOTATION

Set builder notation :

Here we are going to see how to how to represent a set using set builder notation.

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

‘|’or ‘:’ such that

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

“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,

## Set builder notation - Examples

Example 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}.

Example 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 }

Example 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 = 1x = 2nx = 21x = 2 n = 2x = 2nx = 22x = 4 n = 3x = 2nx = 23x = 8 n = 4x = 2nx = 24x = 16 n = 5x = 2nx = 25x = 32

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

Example 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 }

Example 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}

Example 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}

Example 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}

Example 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} After having gone through the stuff given above, we hope that the students would have understood "Set builder notation".

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