Table of Contents
Question 1: Which of the following are sets? Justify your answer.
i. The collection of all months of a year beginning with the letter J.
- Solution: This is a set. The elements are well-defined: January, June, July. We can definitively determine which months belong to this collection and which do not.
ii. The collection of ten most talented writers of India.
- Solution: This is not a set. The term “most talented” is subjective and not well-defined. The criteria for talent can vary from person to person, so a definite list cannot be created.
iii. A team of eleven best cricket batsmen of the world.
- Solution: This is not a set. The term “best” is subjective. The criteria for being the “best” batsman can differ from one person to another, making the collection not well-defined.
iv. The collection of all boys in your class.
- Solution: This is a set. The collection is well-defined. You can easily and definitively identify all the boys in your class.
v. The collection of all natural numbers less than 100.
- Solution: This is a set. The elements are well-defined: {1, 2, 3, …, 99}. It is a countable and definite collection.
vi. A collection of novels written by the writer Munshi Prem Chand.
- Solution: This is a set. The collection is well-defined as all the novels written by Munshi Prem Chand are a specific and identifiable list.
vii. The collection of all even integers.
- Solution: This is a set. The collection is well-defined. An even integer is a definite object (…, -4, -2, 0, 2, 4, …).
viii. The collection of questions in this chapter.
- Solution: This is a set. The collection of questions is well-defined and can be clearly identified within the chapter.
ix. A collection of the most dangerous animals in the world.
- Solution: This is not a set. The term “most dangerous” is subjective and not well-defined. An animal that is considered dangerous to one person may not be to another.
Question 2: Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces.
i. 5 … A
- Solution: Since 5 is an element of set A, the correct symbol is ∈.
- 5 ∈ A
ii. 8 … A
- Solution: Since 8 is not an element of set A, the correct symbol is ∉.
- 8 ∉ A
iii. 0 … A
- Solution: Since 0 is not an element of set A, the correct symbol is ∉.
- 0 ∉ A
iv. 4 … A
- Solution: Since 4 is an element of set A, the correct symbol is ∈.
- 4 ∈ A
v. 2 … A
- Solution: Since 2 is an element of set A, the correct symbol is ∈.
- 2 ∈ A
vi. 10 … A
- Solution: Since 10 is not an element of set A, the correct symbol is ∉.
- 10 ∉ A
Question 3: Write the following sets in roster form.
i. A = {x : x is an integer and -3 ≤ x < 7}
- Solution: The integers from -3 to 6 (inclusive of -3, but not 7) are listed.
- A = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}
ii. B = {x : x is a natural number less than 6}
- Solution: The natural numbers less than 6 are listed.
- B = {1, 2, 3, 4, 5}
iii. C = {x : x is a two-digit natural number such that the sum of its digits is 8}
- Solution: The two-digit numbers whose digits sum to 8 are listed.
- C = {17, 26, 35, 44, 53, 62, 71, 80}
iv. D = {x : x is a prime number which is a divisor of 60}
- Solution: The prime numbers that divide 60 are found and listed. The prime factors of 60 are 2, 3, and 5.
- D = {2, 3, 5}
v. E = The set of all letters in the word TRIGONOMETRY.
- Solution: The unique letters of the word are listed. Note that repeated letters are listed only once.
- E = {T, R, I, G, O, N, M, E, Y}
vi. F = The set of all letters in the word BETTER.
- Solution: The unique letters of the word are listed.
- F = {B, E, T, R}
Question 4: Write the following sets in set-builder form.
i. {3, 6, 9, 12}
- Solution: This is a set of multiples of 3.
- {x : x = 3n, n ∈ N and 1 ≤ n ≤ 4}
ii. {2, 4, 8, 16, 32}
- Solution: This is a set of powers of 2.
- {x : x = 2ⁿ, n ∈ N and 1 ≤ n ≤ 5}
iii. {5, 25, 125, 625}
- Solution: This is a set of powers of 5.
- {x : x = 5ⁿ, n ∈ N and 1 ≤ n ≤ 4}
iv. {2, 4, 6, …}
- Solution: This is an infinite set of even natural numbers.
- {x : x is an even natural number}
v. {1, 4, 9, …, 100}
- Solution: This is a set of squares of natural numbers from 1 to 10.
- {x : x = n², n ∈ N and 1 ≤ n ≤ 10}
Question 5: List all the elements of the following sets.
i. A = {x : x is an odd natural number}
- Solution: The odd natural numbers are listed.
- A = {1, 3, 5, 7, …}
ii. B = {x : x is an integer, -1/2 < x < 9/2}
- Solution: The integers between -0.5 and 4.5 are listed.
- B = {0, 1, 2, 3, 4}
iii. C = {x : x is an integer, x² ≤ 4}
- Solution: The integers whose square is less than or equal to 4 are listed.
- C = {-2, -1, 0, 1, 2}
iv. D = {x : x is a letter in the word “LOYAL”}
- Solution: The unique letters of the word are listed.
- D = {L, O, Y, A}
v. E = {x : x is a month of a year not having 31 days}
- Solution: The months with less than 31 days are listed.
- E = {February, April, June, September, November}
vi. F = {x : x is a consonant in the English alphabet which precedes k}
- Solution: The consonants before k are listed.
- F = {B, C, D, F, G, H, J}
Question 6: Match each of the set on the left in the roster form with the same set on the right in the set-builder form.
i. {1, 2, 3, 6} ii. {2, 3} iii. {M, A, T, H, E, I, C, S} iv. {1, 3, 5, 7, 9}
A. {x : x is a prime number and a divisor of 6} B. {x : x is an odd natural number less than 10} C. {x : x is a natural number and a divisor of 6} D. {x : x is a letter of the word MATHEMATICS}
- Solutions:
- i. {1, 2, 3, 6} matches with C. (Natural numbers that are divisors of 6).
- ii. {2, 3} matches with A. (Prime numbers that are divisors of 6).
- iii. {M, A, T, H, E, I, C, S} matches with D. (Letters in the word MATHEMATICS).
- iv. {1, 3, 5, 7, 9} matches with B. (Odd natural numbers less than 10).