EXERCISE 1.1
SELECT QUESTION
Q1. Which of the following are sets? Justify your answer.
a) The collection of all the months of a year starting with the letter J.This is a set because the months of the year starting with the letter 'J' are well-defined and finite: January, June, and July. Thus, the set is clearly defined and no ambiguity exists.
Answer: It is a set.
b) The collection of ten most talented writers of India. This is not a set because the term "most talented" is subjective. Different people might have different opinions on who the most talented writers are. Hence, it is not a well-defined collection.
Answer: It is not a set.
c) A team of eleven best cricket batsmen of the world. This is not a set because, again, the term "best" is subjective. Different people may have different opinions about who the best cricket batsmen are.
Answer: It is not a set.
d) The collection of all boys in your class. This is a set because the collection of boys in a class is a well-defined group. The number of boys can be counted, and it remains fixed within a particular class.
Answer: It is a set.
e) The collection of all natural numbers less than 100.This is a set because the natural numbers less than 100 are well-defined and finite. We can easily list them as {1, 2, 3, ..., 99}.
Answer: It is a set.
f) The collection of novels written by the writer Munshi Prem Chand. This is a set because the novels written by Munshi Prem Chand can be clearly listed, and the collection is finite.
Answer: It is a set.
g) The collection of all even integers. This is a set because even though the collection of even integers is infinite, it is well-defined. The set of even integers is clear: {..., -4, -2, 0, 2, 4, ...}.
Answer: It is a set.
h) The collection of questions in this chapter. This is a set because the questions in this chapter are finite and well-defined. We can count and list all the questions.
Answer: It is a set.
i ) A collection of most dangerous animals of the world. This is not a set because the term "most dangerous" is subjective. Different people might have different opinions about which animals are the most dangerous.
Answer: It is not a set.
Q2. Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol (∈ or ∉) between the elements and the set.
a) 2 __ ASince 2 is an element of the set A (A = {1, 2, 3, 4, 5, 6}), we write:
Answer: 2 ∈ A
b) 10 __ ASince 10 is not an element of the set A (A = {1, 2, 3, 4, 5, 6}), we write:
Answer: 10 ∉ A
c) 5 __ ASince 5 is an element of the set A (A = {1, 2, 3, 4, 5, 6}), we write:
Answer: 5 ∈ A
d) 0 __ ASince 0 is not an element of the set A (A = {1, 2, 3, 4, 5, 6}), we write:
Answer: 0 ∉ A
Q3. Write the following sets in roster form:a) A = {x: x is an integer and -3 ≤ x < 7}Here, the set A consists of all integers between -3 (inclusive) and 7 (exclusive). The integers in this range are -3, -2, -1, 0, 1, 2, 3, 4, 5, and 6.
Answer: A = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}
b) B = {x: x is a natural number less than 6}The natural numbers are 1, 2, 3, 4, and 5, since natural numbers start from 1 and are less than 6.
Answer:B = {1, 2, 3, 4, 5}
c) C = {x: x is a two-digit natural number such that the sum of its digits is 8}We need to find all two-digit numbers whose digits sum to 8:
- 17 (1 + 7 = 8)
- 26 (2 + 6 = 8)
- 35 (3 + 5 = 8)
- 44 (4 + 4 = 8)
- 53 (5 + 3 = 8)
- 62 (6 + 2 = 8)
- 71 (7 + 1 = 8)
- 80 (8 + 0 = 8)
Answer: C = {17, 26, 35, 44, 53, 62, 71, 80}
d) D = {x: x is a prime number which is a divisor of 60}The divisors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. From this list, the prime divisors are 2, 3, and 5.
Answer:D = {2, 3, 5}
Q4. Write the following sets in set-builder form:
a) A = {2, 4, 6, 8, 10, 12} We observe that each element of the set is an even number. The set consists of all even natural numbers less than or equal to 12.
Answer: A = {x : x is an even natural number, x ≤ 12}
b) B = {3, 6, 9, 12, 15}The elements of the set are multiples of 3. The set consists of all multiples of 3 between 3 and 15.
Answer: B = {x : x is a multiple of 3, 3 ≤ x ≤ 15}
c) C = {2, 3, 5, 7, 11}The elements of the set are prime numbers less than or equal to 11.
Answer: C = {x : x is a prime number, x ≤ 11}
d) D = {1, 4, 9, 16, 25}Each element of the set is a perfect square. These are the squares of the numbers 1, 2, 3, 4, and 5.
Answer: D = {x : x = n², where n is a natural number, 1 ≤ n ≤ 5}
Q5. List all the elements of the following sets:
a) A = {x : x is an odd natural number less than 10} Odd natural numbers less than 10 are 1, 3, 5, 7, 9.
Answer: A = {1, 3, 5, 7, 9}
b) B = {x : x is a prime number less than 20}Prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, 19.
Answer: B = {2, 3, 5, 7, 11, 13, 17, 19}
c) C = {x : x is a natural number which divides 42}The divisors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42.
Answer: C = {1, 2, 3, 6, 7, 14, 21, 42}
d) D = {x : x is a letter in the word “LOYAL”}The distinct letters in the word "LOYAL" are L, O, Y, A.
Answer: D = {L, O, Y, A}
