EXERCISE 1.1
1. Which of the following are sets? Justify your answer.
(i) The collection of all the months of a year beginning with the letter J.
Solution: Yes, this is a set. This collection is well-defined. The months are January, June, July. We can clearly determine whether any given month belongs to this collection or not.
(ii) The collection of ten most talented writers of India.
Solution: No, this is not a set. This collection is not well-defined. The term “most talented” is subjective and can vary from person to person. There is no clear criterion to determine who the “ten most talented” writers are.
(iii) A team of eleven best-cricket batsmen of the world.
Solution: No, this is not a set. Similar to the previous example, “best-cricket batsmen” is a subjective criterion. What one person considers “best,” another might not. Thus, the collection is not well-defined.
(iv) The collection of all boys in your class.
Solution: Yes, this is a set. This collection is well-defined. You can clearly identify every boy in your class, and it’s unambiguous whether a student belongs to this collection or not.
(v) The collection of all natural numbers less than 100.
Solution: Yes, this is a set. This collection is well-defined. Natural numbers are clearly defined (1, 2, 3, …), and “less than 100” provides a clear boundary. We can definitively say if a number is in this collection (e.g., 50 is, 101 is not).
(vi) A collection of novels written by the writer Munshi Prem Chand.
Solution: Yes, this is a set. This collection is well-defined. We can definitively determine whether a particular novel was written by Munshi Prem Chand or not.
(vii) The collection of all even integers.
Solution: Yes, this is a set. This collection is well-defined. Even integers are clearly defined (…, -4, -2, 0, 2, 4, …). We can definitively say if any integer is even or not.
(viii) The collection of questions in this Chapter.
Solution: Yes, this is a set. This collection is well-defined. You can clearly identify all the questions present in this specific chapter.
(ix) A collection of most dangerous animals of the world.
Solution: No, this is not a set. The term “most dangerous” is subjective. Danger can be defined in various ways (e.g., to humans, to other animals, by venom, by size). Therefore, this collection is not well-defined.
2. Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces:
(i) 5 ___ A
Solution: 5 ∈ A (5 is an element of A)
(ii) 8 ___ A
Solution: 8 ∉ A (8 is not an element of A)
(iii) 0 ___ A
Solution: 0 ∉ A (0 is not an element of A)
(iv) 4 ___ A
Solution: 4 ∈ A (4 is an element of A)
(v) 2 ___ A
Solution: 2 ∈ A (2 is an element of A)
(vi) 10 ___ A
Solution: 10 ∉ A (10 is not an element of A)
3. Write the following sets in roster form:
(i) A = {x : x is an integer and –3 ≤ x < 7}
Solution: A = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}
(ii) B = {x : x is a natural number less than 6}
Solution: 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: C = {17, 26, 35, 44, 53, 62, 71, 80}
(iv) D = {x : x is a prime number which is divisor of 60}
Solution: The divisors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. Among these, the prime divisors are 2, 3, 5. Therefore, D = {2, 3, 5}
(v) E = The set of all letters in the word TRIGONOMETRY
Solution: E = {T, R, I, G, O, N, M, E, Y} (Note: Duplicate letters are listed only once)
(vi) F = The set of all letters in the word BETTER
Solution: F = {B, E, T, R}
4. Write the following sets in the set-builder form:
(i) {3, 6, 9, 12}
Solution: {x : x = 3n, where n ∈ N and 1 ≤ n ≤ 4} Alternatively: {x : x is a multiple of 3 and 3 ≤ x ≤ 12}
(ii) {2, 4, 8, 16, 32}
Solution: {x : x = 2ⁿ, where n ∈ N and 1 ≤ n ≤ 5}
(iii) {5, 25, 125, 625}
Solution: {x : x = 5ⁿ, where n ∈ N and 1 ≤ n ≤ 4}
(iv) {2, 4, 6, . . .}
Solution: {x : x is an even natural number} Alternatively: {x : x = 2n, where n ∈ N}
(v) {1, 4, 9, . . ., 100}
Solution: {x : x = n², where n ∈ N and 1 ≤ n ≤ 10}
5. List all the elements of the following sets:
(i) A = {x : x is an odd natural number}
Solution: A = {1, 3, 5, 7, …} (This is an infinite set)
(ii) B = {x : x is an integer, –½ < x < 9/2}
Solution: Converting to decimals: -0.5 < x < 4.5. The integers in this range are 0, 1, 2, 3, 4. Therefore, B = {0, 1, 2, 3, 4}
(iii) C = {x : x is an integer, x² ≤ 4}
Solution: We need to find integers where x² ≤ 4. Testing values: if x = 0, then 0² = 0 ≤ 4 (True); if x = 1, then 1² = 1 ≤ 4 (True); if x = -1, then (-1)² = 1 ≤ 4 (True); if x = 2, then 2² = 4 ≤ 4 (True); if x = -2, then (-2)² = 4 ≤ 4 (True); if x = 3, then 3² = 9 ≤ 4 (False). Therefore, C = {-2, -1, 0, 1, 2}
(iv) D = {x : x is a letter in the word “LOYAL”}
Solution: D = {L, O, Y, A}
(v) E = {x : x is a month of a year not having 31 days}
Solution: The months with 31 days are January, March, May, July, August, October, December. The months without 31 days are February, April, June, September, November. Therefore, E = {February, April, June, September, November}
(vi) F = {x : x is a consonant in the English alphabet which precedes k}
Solution: The English alphabet up to k includes A, B, C, D, E, F, G, H, I, J, K. The consonants in this range are B, C, D, F, G, H, J. Therefore, F = {B, C, D, F, G, H, J}
6. Match each of the set on the left in the roster form with the same set on the right described in set-builder form:
(i) {1, 2, 3, 6} ↔ (c) {x : x is natural number and divisor of 6}
(ii) {2, 3} ↔ (a) {x : x is a prime number and a divisor of 6}
(iii) {M, A, T, H, E, I, C, S} ↔ (d) {x : x is a letter of the word MATHEMATICS}
(iv) {1, 3, 5, 7, 9} ↔ (b) {x : x is an odd natural number less than 10}
