**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.

**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,

**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 = 2^{n}, n ∈ N and n ≤ 5}

**Solution :**

X = {x : x = 2^{n}, 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 = 2 x = 2 x = 2 |
n = 2 x = 2 x = 2 x = 4 |
n = 3 x = 2 x = 2 x = 8 |
n = 4 x = 2 x = 2 x = 16 |
n = 5 x = 2 x = 2 x = 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".

Apart from the stuff given above, if you want to know more about "Set builder notation", please click here

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