Roster Form and Set-Builder Form of Sets

A set is a collection of well-defined objects. These objects may be actually listed or may be specified by a rule. In this article, we shall study the application of the definition of a set. Similarly, we shall study to write sets by roster method and set-builder method.

Problems on Definition of a Set:

Which of the following collections are set?

• The collection of easy topics of mathematics

The term used ‘easy topic’ is a relative term and choice may vary from person to person. Hence it is not a collection of well-defined objects. It is not a set.

• The collection of even natural numbers

It is a collection of well-defined objects. It is a set.

• The collection of rich people in the world

The term used ‘rich people’ is a relative term and choice may vary from person to person. Hence it is not a collection of well-defined objects. It is not a set.

• The collection of all books in Asiatic Library

It is a collection of well-defined objects. It is a set.

• The collection of clever students of your class

The term used ‘clever students in your class’ is a relative term and choice may vary from person to person. Hence it is not a collection of well-defined objects. It is not a set.

• The collection of numbers divisible by 3

It is a collection of well-defined objects. It is a set.

• a, e, i, o, u

It is a collection of well-defined objects. It is a set.

• The collection of English alphabet

It is a collection of well-defined objects. It is a set.

• The collection of happy persons in your country

The term used ‘happy person in your country’ is a relative term and choice may vary from person to person. Hence it is not a collection of well-defined objects. It is not a set.

• The collection of diagrams in your science book

It is a collection of well-defined objects. It is a set.

• 1/2, 1/3, 1/4, 1/5, 1/6

It is a collection of well-defined objects. It is a set.

• 2, 4, 8, 16, 32

It is a collection of well-defined objects. It is a set.

• The collection of the best TV channels

The term used ‘best TV channels’ is a relative term and choice may vary from person to person. Hence it is not a collection of well-defined objects. It is not a set.

• The collection of colours of rainbow

It is a collection of well-defined objects. It is a set.

• The collection of tasty foods

The term used ‘tasty foods’ is a relative term and choice may vary from person to person. Hence it is not a collection of well-defined objects. It is not a set.

• The collection of odd positive integers

It is a collection of well-defined objects. It is a set.

• The collection of poor people in Africa

The term used ‘poor people’ is a relative term and choice may vary from person to person. Hence it is not a collection of well-defined objects. It is not a set.

• The collection of whole numbers less than 10

It is a collection of well-defined objects. It is a set.

• Collections of vowels of English alphabet

It is a collection of well-defined objects. It is a set.

• A Collection of numbers multiple of 7

It is a collection of well-defined objects. It is a set.

• A collection of beautiful girls in your area

The term used ‘beautiful girls’ is a relative term and choice may vary from person to person. Hence it is not a collection of well-defined objects. It is not a set.

• A collection of integers

It is a collection of well-defined objects. It is a set.

Problems on Writing Set in Roster Form:

Write the following sets in Roster Form.

• A = {x| x ∈ I, -3 ≤ x ≤ 3}

Set in roster form is A = {-3, -2, -1, 0, 1, 2, 3}

• B = {x| x ∈ I, x ∉ W}

Set in roster form is B = {….., -3, -2, -1}

• C = {x| x ∈ I, x ∉ N}

Set in roster form is C = {….., -3, -2, -1, 0}

• D = {x| x ∈ I, x² < 10}

Set in roster form is D = {-3, -2, -1, 0, 1, 2, 3}

• E = {x| x ∈ I, -3 < x ≤ 3}

Set in roster form is A = {-2, -1, 0, 1, 2, 3}

• F = {x| x = (n/(n²-1)), n ∈ N, 2 < x ≤ 4}

n can take values 3 and 4

When n = 3, x = (n/(n²-1)) = (3/(3²-1)) = 3/8

When n = 4, x = (n/(n²-1)) = (4/(4²-1)) = 4/15

Set in roster form is F = {3/8, 4/15}

• G = {x| x is odd prime number less than 10}

Set in roster form is G = {2, 3, 5, 7}

Note: 1 is not a prime number

• H = {x| x is even multiple of 5 less than 60}

Set in roster form is H = {10, 20, 30, 40, 50}

• J = {x| x ∈ W, x ∉ N}

Set in roster form is J = {0}

• K = {x| x is types of triangle based on lengths of sides of triangle}

Set in roster form is K = {Equilateral triangle, Isosceles triangle, Scalene triangle}

• L = {x| x is types of triangle based on measures of angles of triangle}

Set in roster form is L = {Acute angled triangle, Right-angled triangle, Obtuse angled triangle}

• M = {x| x is a vowel of English Alphabet}

Set in roster form is M = {a, e, i, o, u}

• P = {x| x isgreatest two digit, three digit, and four digit number}

Set in roster form is P = {99, 999, 9999}

• S = {x| x ∈ N, 3x – 1 < 8}

When x = 1, 3x – 1 = 3(1) – 1 = 2 < 8, 1 ∈ P

When x = 2, 3x – 1 = 3(2) – 1 = 5 < 8, 2 ∈ P

When x = 3, 3x – 1 = 3(3) – 1 = 8 , 3 ∉ P

Set in roster form is S = {1, 2}

• T = {x| x is divisor of 24}

Set in roster form is T = {1, 2, 3, 4, 6, 8, 12, 24}

• V = {x| x is prime divisor of 24}

Set in roster form is V = {2, 3}

• Y = {x| x is a vowel in the word LOGARITHM}

Set in roster form is Y = {O, A, I}

• A = {x| x ∈ I, x² – 9 = 0}

x² – 9 = 0

x² = 9

x = 3 or x = -3

Set in roster form is A = {-3, 3}

• B = {x| x is non negative integer}

Set in roster form is B = {0, 1, 2, 3, ……….}

• C = {x| x is two-digit natural number such that the sum of its digit is 7}

Set in roster form isC = {16, 25, 34, 43, 52, 61, 70}

• D = {x| x is a square of prime number less than 10}

Set in roster form is D = {4, 9, 25, 49}

Problems on Writing Set in Set Builder Form:

Write the following sets in the Set-Builder Form.

• A = {9, 16, 25, 36, …..,81}

Set-builder form is A = {x| x = n², n ∈ N, 3 ≤ n ≤ 9}

• B = {8, – 8}

Set-builder form is B = {x| x ∈ I, x² – 64 = 0}

• C = {15, 24, 33, 42, 51, 60}

Set-builder form is C = {x| x is two digi natural number whose sum of digits is 6}

• D = {-4, 4}

Set-builder form is D = {x| x ∈ I, x² – 16 = 0}

• E = {1, 8, 27, 64, 125}

Set-builder form is E = {x| x = n³, n ∈ N, 1 ≤ n ≤ 5}

• F = {3, 9, 15, 21, 27, …….}

Set-builder form is F = {x| x = 3n, n is odd natural number}

• G = {1/2, 2/5, 3, 10, 4/17, 5/26}

Set-builder form is G = {x| x = (n/(n² + 1)), n ∈ N, 1 ≤ n ≤ 5}

• H = {a, a + 2, a + 4, a + 6, …………..}

Set-builder form is H = {x| x = a + 2n, n ∈ W}

J = {1, 2, 3, 4, 6, 12}

Set-builder form is J = {x| x is divisor of 12}

• K = {x – 3, x – 6, x – 9, x – 12, …………..}

Set-builder form is K = {x| x = x – 3n, n ∈ N}

• L = {2, 5, 8, 11, 14, ……….}

Set-builder form is L = {x| x = 3n – 1, n ∈ N}

• M = {17, 26, 35, 44, 53, 62, 71, 80}

Set-builder form is M = {x| x is two digit natural number whose sum of digits is 8}

• N = {2, 3, 5, 7, 11, 13}

Set-builder form is N = {x| x is natural prime mumber less than 15}

• P = {1, 2, 3, 6, 9, 18}

Set-builder form is P = {x| x is divisor of 18}

• Q = {5, 10, 15, 20, 25}

Set-builder form is Q = {x| x = 5n, n ∈ N, 1 ≤ n ≤ 5}

• S = {31, 33, 35, 37, 39}

Set-builder form is S = {x| x is odd number, x ∈ N, 30 ≤ x ≤ 40}

• T = {-2, -1, 0, 1, 2}

Set-builder form is T = {x| x ∈ I, -2 ≤ x ≤ 2}

• V = {2³, 2⁴, 2⁵, ……..}

Set-builder form is V = {x| x = 2ⁿ, n ∈ N, n ≥ 3}

• Y = {3², 3³, 3⁴, ……..}

Set-builder form is Y = {x| x = 3ⁿ, n ∈ N, n ≥ 2}