**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".

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