Representation of a Function in set-builder form

A set is a well-defined group of objects or numbers. In maths, a set can be illustrated in several ways. As per the set theory, a set can be described in three ways: descriptive form, roster form, and set builder form. In this article, we will discuss the set builder form.

What is Set Builder form?

In the set builder form the real components of the sets are not listed inside a pair of curly braces but rather an assertion or formula is mentioned inside the brackets. 

If we take example of P that has a set of numbers that are greater than 10, then, 

P={x:x is a number greater than 10}

Or it can be written as

P= {x|x is a counting number greater than 10}

Solved Questions On Set Builder Form

Question 1: Write the given sets in set builder form 

i) {2, 4, 6, 8, 10} 

(ii) {2, 3, 5, 7, 11} 

(iii) {January, June, July}

(iv) {a, e, i, o, u}

(v) {Tuesday, Thursday}

(vi) {1, 4, 9, 16, 25}

(vii) {5, 10, 15, 20, 25, 30}

Solution

The mentioned sets can be illustrated in set builder form as given below:

1. {x: x is less than 12 and even natural number}

2. {x: x is less than 12 and a prime number}

3. {x: x starts from j and is a month name}

4. { x: x is an english alphabet and a vowel}

5. {x: x starts with T and is name of week name}

6. {x: x is a number less than 25 and a perfect square}

7. {x: x divisible by 5 and is a natural number less than 30}

Question 2: Convert the given 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 these elements can be given as 1, 2, 3, 4, ……

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

Question 3: Convert the following sets into set builder form. 

 I) {1/4, 2/5, 3/6, 4/7, 5/8}

 II) { …., -5, 0, 5, 10, ….} 

III) {-4, 4} 

Solution

1. {1/4, 2/5, 3/6, 4/7, 5/8}

You can write set builder form for this article as 

{x : x = n/(n+3), n is a natural number and 1 ≤ n ≤ 5}

2. { …., -5, 0, 5, 10, ….}

You can write the set builder form as mentioned below.

{x: x is a multiple of 5 and x is an integer} 

3. {-4, 4}

Converting to set builder form can be given as: 

{x : x is an integer and x² – 16 = 0} 

Question 4: Convert each of the given sets in set builder form and roster form. 

(i) natural numbers set which are divisible by 30.

(ii) odd number set between 36 and 45.

(iii) even natural number sets which are greater than 24 but less than 40.

(iv) Set of letters used in the word ‘MASSACHUSETTS’.

(v) Set containing names of the year’s last five months.

Solution

(i)Roster Form: {30,15,10,6,5,3,2,1};

Set-Builder Form: {x : x is a natural number all the divisors of 30}

(ii) Roster Form: {37,39,41,43};

Set-Builder Form: {x : x is all natural numbers between 36 and 45}.

(iii) Roster Form: {26,28,30,32,34,36,38};

Set-Builder Form: {x: x is an even natural number greater than 25 but less than 40}.

(iv) Roster Form: {m, a, s, c, h, u, e, t};

Set-Builder Form: {x: x is a letter used in the word ‘MASSACHUSETTS’}.

(v) Roster Form: {august,september,october,november,december};

Set-Builder Form: {x: x is the name of the last five months of a year}

(vi) Roster Form: {16, 25, 36, 49, 64, 81};

Set-Builder Form: {x: x is a perfect square two-digit number}

(vi) Roster Form: {e, d, u, c, a, t, i, o, n};

Set-Builder Form: {x: x is a letter used in the word ‘EDUCATION’}.

Question 5: Write the given sets in set builder form.

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

Solution

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

The given set contains each day of the week. 

So, you can write the set builder form as following: 

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

Conclusion

Set builder form is described as an interpretation or notation that can be used to describe a set defined by a logical formula that simplifies to true for a set element. There may be one or more variables in the set builder notation. It also specifies a rule for determining which elements do not belong to the set and which do. This article describes the representation of a function in a set builder form.