Chapter 1 Sets : Exercise
Math : Click Here
1. Which of the following are sets ? Justify your answer.
Note: A set is a well-defined collection of objects. We must be able to clearly decide whether an object belongs to the collection or not.
(i) The collection of all the months of a year beginning with the letter J.
Ans: Yes, it is a set.
Reason: The months beginning with J are clearly identifiable: {January, June, July}. It is a well-defined collection.
(ii) The collection of ten most talented writers of India.
Ans: No, it is not a set.
Reason: “Most talented” is subjective. Different people may choose different writers. Therefore, it is not well-defined.
(iii) A team of eleven best cricket batsmen of the world.
Ans: No, it is not a set.
Reason: “Best batsmen” is not clearly defined. Different people may select different players.
(iv) The collection of all boys in your class.
Ans: Yes, it is a set.
Reason: We can clearly identify whether a particular student is a boy in the class or not. Hence, it is well-defined.
(v) The collection of all natural numbers less than 100.
Ans: Yes, it is a set.
Reason: The collection is clearly defined.{1, 2, 3, ..., 99}
(vi) A collection of novels written by the writer Munshi Prem Chand.
Ans: Yes, it is a set.
Reason: We can clearly determine which novels were written by Munshi Prem Chand. Therefore, it is well-defined.
(vii) The collection of all even integers.
Ans: Yes, it is a set.
Reason: Even integers are clearly defined as integers divisible by 2.
Examples: {..., −6, −4, −2, 0, 2, 4, 6, ...}
(viii) The collection of questions in this Chapter.
Ans: Yes, it is a set.
Reason: The questions in the chapter are clearly identifiable. Hence, it is a well-defined collection.
(ix) A collection of most dangerous animals of the world.
Ans: No, it is not a set.
Reason: “Most dangerous” is not precisely defined. Different people may have different opinions about which animals are the most dangerous.
========================================================
2. Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces: (i) 5. . .A (ii) 8 . . . A (iii) 0. . .A (iv) 4. . . A (v) 2. . .A (vi) 10. . .A
Ans: (i) 5 ∈ A (ii) 8 ∉ A (iii) 0 ∉ A (iv) 4 ∈ A (v) 2 ∈ A (vi) 10 ∉ A
===================================================
3. Write the following sets in Roster Form
(i) A = {x : x is an integer and −3 ≤ x < 7}
We need all integers from −3 to 6.
A = {−3, −2, −1, 0, 1, 2, 3, 4, 5, 6}
(ii) B = {x : x is a natural number less than 6}
Natural numbers less than 6 are: 1, 2, 3, 4, 5
Therefore, 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}
Two-digit numbers whose digits add up to 8 are: 17, 26, 35, 44, 53, 62, 71, 80
Therefore, C = {17, 26, 35, 44, 53, 62, 71, 80}
(iv) D = {x : x is a prime number which is a divisor of 60}
Prime factors of 60: 60 = 2² × 3 × 5
Therefore, the prime divisors are: 2, 3, 5
Hence, D = {2, 3, 5}
(v) E = The set of all letters in the word TRIGONOMETRY
Word: TRIGONOMETRY
The letters are: T, R, I, G, O, N, O, M, E, T, R, Y
Repeated letters are written only once.
Therefore,
E = {T, R, I, G, O, N, M, E, Y}
(vi) F = The set of all letters in the word BETTER
Word: BETTER
Letters: B, E, T, T, E, R
Since repeated letters are written only once: F = {B, E, T, R}
Math : Click Here
==============================================
4. Write the following sets in Set-Builder Form
(i) {3, 6, 9, 12}
These are the positive multiples of 3 up to 12.
Ans: A = {x : x = 3n, n ∈ N, 1 ≤ n ≤4}
(ii) {2, 4, 8, 16, 32}
These are powers of 2: 2¹, 2², 2³, 2⁴, 2⁵
Ans: B = {x : x = 2n, n ∈ N, 1 ≤ n ≤5}
(iii) {5, 25, 125, 625}
These are powers of 5: 5¹, 5², 5³, 5⁴
Ans: C = {x : x = 5n, n ∈ N, 1 ≤ n ≤4}
(iv) {2, 4, 6, ...}
These are all positive even natural numbers.
Ans: D = {x : x = 2n, n ∈ N}
(v) {1, 4, 9, ..., 100}
These are the squares of natural numbers from 1² to 10².
Ans: E = {x : x = n2, n ∈ N, 1 ≤ n ≤10}
Trick
- 3, 6, 9, 12 → multiples of 3
- 2, 4, 8, 16, 32 → powers of 2
- 5, 25, 125, 625 → powers of 5
- 2, 4, 6, ... → even numbers
- 1, 4, 9, ..., 100 → square numbers
===============================================
5. List All the Elements of the Following Sets
(i) A = {x : x is an odd natural number}
Odd natural numbers are: 1, 3, 5, 7, 9, ...
Therefore, A = {1,3,5,7,9,…}
(ii) B = {x : x is an integer, −1/2 < x < 9/2}
We need integers between −0.5 and 4.5.
The integers are: 0, 1, 2, 3, 4
Therefore, B = {0,1,2,3,4}
(iii) C = {x : x is an integer, x² ≤ 4}
We have: x2 ≤ 4
So,
−2 ≤ x ≤ 2
The integers are: −2, −1, 0, 1, 2
Therefore, C = {−2, −1, 0, 1, 2}
(iv) D = {x : x is a letter in the word “LOYAL”}
The letters are: L, O, Y, A, L
Since L is repeated, we write it only once.
Therefore, D = {L,O,Y,A}
(v) E = {x : x is a month of a year not having 31 days}
Months that do not have 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}
The letters before k are: a, b, c, d, e, f, g, h, i, j
Among these, the consonants are: b, c, d, f, g, h, j
Therefore, F = {b, c, d, f, g, h, j}
Ans: (i) A = {1, 3, 5, 7, 9, ...}, (ii) B = {0, 1, 2, 3, 4}, (iii) C = {−2, −1, 0, 1, 2}, (iv) D = {L, O, Y, A}, (v) E = {February, April, June, September, November}, (vi) F = {b, c, d, f, g, h, j}
===================================
Q6. 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} (a) {x : x is a prime number and a divisor of 6}
(ii) {2, 3} (b) {x : x is an odd natural number less than 10}
(iii) {M,A,T,H,E,I,C,S} (c) {x : x is natural number and divisor of 6}
(iv) {1, 3, 5, 7, 9} (d) {x : x is a letter of the word MATHEMATICS}
Answer:
(i) {1, 2, 3, 6} : These are the natural numbers that are divisors of 6.
Ans: (i) → (c)
(ii) {2, 3} : These are the prime numbers that are divisors of 6.
Ans: (ii) → (a)
(iii) {M, A, T, H, E, I, C, S} : These are the letters of the word MATHEMATICS. Repeated letters are written only once.
Ans: (iii) → (d)
(iv) {1, 3, 5, 7, 9} : These are the odd natural numbers less than 10.
Ans: (iv) → (b)
=====================================@
Math : Click Here