NCERT Solutions for Class 11 Maths Miscellaneous Exercise Chapter 1 - Sets provide a comprehensive review of the concepts covered in the chapter on Sets. This exercise helps students consolidate their understanding of set operations, including union, intersection, difference, and complement of sets, as well as Venn diagrams and types of sets. It includes a variety of questions that test the application of set theory in different contexts.
This exercise is essential for preparing for exams as it allows students to practice applying the concepts in various problem-solving situations and reinforces the methods of representing and operating with sets.
NCERT Solutions For Class 11 Maths Miscellaneous Exercise Chapter 1 Sets
Q.
Which of the following are sets ? Justify your answer.
(i) The collection of all the months of a year beginning with the letter J.
(ii) The collection of ten most talented writers of India.
(iii) A team of eleven best-cricket batsmen of the world.
(iv) The collection of all boys in your class.
(v) The collection of all natural numbers less than 100.
(vi) A collection of novels written by the writer Munshi Prem Chand.
(vii) The collection of all even integers.
(viii) The collection of questions in this Chapter.
(ix) A collection of most dangerous animals of the world.
Q.
In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers.
Find:
(i) the number of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper.
Q.
Draw appropriate Venn diagram for each of the following :
(i) (A ∪ B)’, (ii) A′ ∩ B′,
(iii) (A ∩ B)′, (iv) A′ ∪ B′
Q.
If X and Y are two sets such that X ∪ Y has 18 elements, X has 8 elements and Y has 15 elements; how many elements does X ∩ Y have?
Q.
In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?
Q.
Decide, among the following sets, which sets are subsets of one and another:
A = {x: x ∈ R and x satisfy x2 – 8x + 12 = 0},
B = {2, 4, 6}, C = {2, 4, 6, 8, . . .}, D = { 6 }.
Q.
In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.
Q.
Assume that P (A) = P (B). Show that A = B
Q.
Using properties of sets, show that
(i) A ∪ ( A ∩ B ) = A
(ii) A ∩ ( A ∪ B ) = A.
Q.
In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many students were taking neither tea nor coffee?
Q.
In a group of students, 100 students know Hindi, 50 know English and 25 know both.
Each of the students knows either Hindi or English. How many students are there in the group?
Q.
In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.
Q.
Taking the set of natural numbers as the universal set, write down the complements of the following sets:
(i) {x: x is an even natural number}
(ii) {x: x is an odd natural number}
(iii) {x: x is a positive multiple of 3}
(iv) {x: x is a prime number}
(v) {x: x is a natural number divisible by 3 and 5}
(vi) {x: x is a perfect square}
(vii) {x: x is a perfect cube}
(viii) {x: x + 5 = 8}
(ix) {x: 2x + 5 = 9}
(x) { x : x ≥ 7 }
(xi) { x : x ∈ N and 2x + 1 > 10 }
Q.
Find the union of each of the following pairs of sets:
(i) X = {1, 3, 5} Y = {1, 2, 3}
(ii) A = [ a, e, i, o, u} B = {a, b, c}
(iii) A = {x: x is a natural number and multiple of 3}
B = {x: x is a natural number less than 6}
(iv) A = {x: x is a natural number and 1 < x ≤ 6}
B = {x: x is a natural number and 6 < x < 10}
(v) A = {1, 2, 3}, B = Φ
Q.
Find the intersection of each pair of sets of
(i) X = {1, 3, 5} Y = {1, 2, 3}
(ii) A = [ a, e, i, o, u} B = {a, b, c}
(iii) A = {x: x is a natural number and multiple of 3}
B = {x: x is a natural number less than 6}
(iv) A = {x: x is a natural number and 1 < x ≤ 6}
B = {x: x is a natural number and 6 < x < 10}
(v) A = {1, 2, 3}, B = Φ
Q.
If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find
(i) A ∩ B
(ii) B ∩ C
(iii) A ∩ C ∩ D
(iv) A ∩ C
(v) B ∩ D
(vi) A ∩ (B ∪ C)
(vii) A ∩ D
(viii) A ∩ (B ∪ D)
(ix) (A ∩ B) ∩ (B ∪ C)
(x) (A ∪ D) ∩ (B ∪ C)
Q.
If A = {x: x is a natural number },
B = {x: x is an even natural number}
C = {x: x is an odd natural number} and
D = {x: x is a prime number}, find
(i) A ∩ B
(ii) A ∩ C
(iii) A ∩ D
(iv) B ∩ C
(v) B ∩ D
(vi) C ∩ D
Q.
Which of the following pairs of sets are disjoint
(i) {1, 2, 3, 4} and {x: x is a natural number and 4 ≤ x ≤ 6 }
(ii) {a, e, i, o, u} and { c, d, e, f}
(iii) {x: x is an even integer} and {x: x is an odd integer}
Q.
State whether each of the following statement is true or false. Justify your answer.
(i) {2, 3, 4, 5} and {3, 6} are disjoint sets.
(ii) {a, e, i, o, u } and { a, b, c, d }are disjoint sets.
(iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.
(iv) {2, 6, 10} and {3, 7, 11} are disjoint sets.
Q.
Let A, B, and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. Show that B = C
Q.
Show that the following four conditions are equivalent:
(i) A ⊂ B
(ii) A – B = Φ
(iii) A ∪ B = B
(iv) A ∩ B = A
Q.
Show that if A ⊂ B, then C – B ⊂ C – A.
Q.
Is it true that for any sets A and B, P (A )∪ P (B) = P (A ∪ B)? Justify your answer.
Q.
If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that
(i) (A ∪ B)′ = A′ ∩ B′
(ii) (A ∩ B)′ = A′ ∪ B′
Q.
If U = {a, b, c, d, e, f, g, h}, find the complements of the following sets:
(i) A = {a, b, c}
(ii) B = {d, e, f, g}
(iii) C = {a, c, e, g}
(iv) D = {f, g, h, a}
Q.
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
Q.
Make correct statements by filling in symbols ⊂ or in the blank spaces:
Q.
Write the following sets in roster form:
(i) A = {x: x is an integer and –3 < x < 7}
(ii) B = {x: x is a natural number less than 6}
(iii) C = {x: x is a two-digit natural number such that the sum of its digits is 8}
(iv) D = {x: x is a prime number which is divisor of 60}
(v) E = The set of all letters in the word TRIGONOMETRY
(vi) F = The set of all letters in the word BETTER
Q.
Write the following sets in the set-builder form:
(i) (3, 6, 9, 12}
(ii) {2, 4, 8,16,32}
(iii) {5, 25, 125, 625}
(iv) {2, 4, 6, . . .}
(v) {1,4,9, . . .,100}
Q.
List all the elements of the following sets:(i)A = x : x is an odd natural number(ii)B = x : x is an integer, –21 < x < 29(iii)C = x : x is an integer, x2≤4(iv)D = x : x is a letter in the word “LOYAL”(v)E = x : x is a month of a year not having 31 days(vi)F = x : x is a consonant in the English alphabet which precedes k.
Q.
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}.
Q.
Which of the following are examples of the null set
(i) Set of odd natural numbers divisible by 2
(ii) Set of even prime numbers
(iii) {x: x is a natural numbers, x < 5 and x > 7 }
(iv) {y : y is a point common to any two parallel lines}
Q.
Which of the following sets are finite or infinite
(i) The set of months of a year
(ii) {1, 2, 3, . . .}
(iii) {1, 2, 3, . . .99, 100}
(iv) The set of positive integers greater than 100
(v) The set of prime numbers less than 99
Q.
State whether each of the following set is finite or infinite:
(i) The set of lines which are parallel to the x-axis
(ii) The set of letters in the English alphabet
(iii) The set of numbers which are multiple of 5
(iv) The set of animals living on the earth
(v) The set of circles passing through the origin (0, 0)
Q.
In the following, state whether A = B or not:
(i) A = {a, b, c, d}; B = { d, c, b, a }
(ii) A = {4, 8, 12, 16}; B = { 8, 4, 16, 18}
(iii) A = {2, 4, 6, 8, 10}; B = {x: x is positive even integer and x ≤ 10}
(iv) A = { x : x is a multiple of 10};
B = { 10, 15, 20, 25, 30, . . . }
Q.
Are the following pair of sets equal? Give reasons.
(i) A = {2, 3},
B = {x: x is solution of x2 + 5x + 6 = 0}
(ii) A = {x: x is a letter in the word FOLLOW}
B = {y: y is a letter in the word WOLF}
Q.
From the sets given below, select equal sets:
A = {2, 4, 8, 12},
B = {1, 2, 3, 4},
C = {4, 8, 12, 14},
D = {3, 1, 4, 2}
E = {–1, 1},
F = {0, a},
G = {1, –1},
H = {0, 1}
Q.
Examine whether the following statements are true or false:(i) a,b⊂b,c, a(ii) a,e⊂x : x is a vowel in the English alphabet(iii) { 1, 2, 3 }⊂ 1, 3,5 (iv) a⊂a,b, c(v) a∈a,b, c(vi) {x : x is an even natural number less than 6}⊂{x : x is a natural number which divides 36}
Q.
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9},
A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find
(i) A′
(ii) B′
(iii) (A ∪ C)′
(iv) (A ∪ B)′
(v) (A′)′
(vi) (B – C)’
Q.
Let A = {1, 2, {3, 4}, 5}. Which of the following statements are incorrect and why?
(i) {3, 4} ⊂ A
(ii) {3, 4} ∈ A
(iii) {{3, 4}} ⊂ A
(iv) 1 ∈ A
(v) 1 ⊂ A
(vi) {1, 2, 5} ⊂ A
(vii) {1, 2, 5} ∈ A
(viii) {1, 2, 3} ⊂ A
(ix) Φ ∈ A
(x) Φ ⊂ A
(xi) {Φ} ⊂ A
Q.
Write down all the subsets of the following sets
(i) {a}
(ii) {a, b}
(iii) {1, 2, 3}
(iv) Φ
Q.
Write the following as intervals:
(i) {x: x ∈ R, – 4 < x ≤ 6}
(ii) {x: x ∈ R, – 12 < x < –10}
(iii) {x: x ∈ R, 0 ≤ x < 7}
(iv) {x: x ∈ R, 3 ≤ x ≤ 4}
Q.
Write the following intervals in set-builder form:
(i) (– 3, 0)
(ii) [6, 12]
(iii) (6, 12]
(iv) [–23, 5)
Q.
What universal set(s) would you propose for each of the following:
(i) The set of right triangles.
(ii) The set of isosceles triangles.
Q.
Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universal set (s) for all the three sets A, B and C
(i) {0, 1, 2, 3, 4, 5, 6}
(ii) Φ
(iii) {0,1,2,3,4,5,6,7,8,9,10}
(iv) {1, 2, 3, 4, 5, 6, 7, 8}
Q.
If A and B are two sets such that A ⊂ B, then what is A ∪ B?
Q.
If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find
(i) A ∪ B
(ii) A ∪ C
(iii) B ∪ C
(iv) B ∪ D
(v) A ∪ B ∪ C
(vi) A ∪ B ∪ D
(vii) B ∪ C ∪ D
Q.
If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}; find
(i) A – B
(ii) A – C
(iii) A – D
(iv) B – A
(v) C – A
(vi) D – A
(vii) B – C
(viii) B – D
(ix) C – B
(x) D – B
(xi) C – D
(xii) D – C
Q.
If R is the set of real numbers and Q is the set of rational numbers, then what is R – Q?
Q.
Let A and B be sets. If A ∩ X = B ∩ X = Φ and A ∪ X = B ∪ X for some set X, show that A = B.
NCERT Solutions For Class 11 Maths Miscellaneous Exercise Chapter 1 Sets
Question 1: Identify Subsets
Decide among the following sets, which sets are the subsets of one and another:
- Set A: {x : x is a real number satisfying x squared - 8x + 12 = 0}
- Set B: {2, 4, 6}
- Set C: {2, 4, 6, 8, ...}
- Set D: {6}
Answer:
- Solving the equation x squared - 8x + 12 = 0 gives (x - 2)(x - 6) = 0, so x = 2 or 6.
- Therefore, Set A = {2, 6}.
- We can observe that:
- D is a subset of A
- A is a subset of B
- B is a subset of C
- The final relationship is: D is a subset of A, which is a subset of B, which is a subset of C (D ⊂ A ⊂ B ⊂ C).
Question 2: True or False Statements
Determine whether each statement is true or false. Prove if true, or give an example if false.
- If x is in A and A is in B, then x is in B:
- Answer: False.
- Example: Let A = {1, 2} and B = {1, {1, 2}, 3}. Here, 2 is in A and A is in B, but 2 is not in B.
- If A is a subset of B and B is in C, then A is in C:
- Answer: False.
- Example: Let A = {2}, B = {0, 2}, and C = {1, {0, 2}, 3}. Here, A is a subset of B and B is in C, but A is not in C.
- If A is a subset of B and B is a subset of C, then A is a subset of C:
- Answer: True.
- Proof: Let x be an element of A. Since A is a subset of B, x must be in B. Since B is a subset of C, x must also be in C.
- If A is not a subset of B and B is not a subset of C, then A is not a subset of C:
- Answer: False.
- Example: Let A = {1, 2}, B = {0, 6, 8}, and C = {0, 1, 2, 6, 9}. Here A is not in B and B is not in C, but A is a subset of C.
- If x is in A and A is not a subset of B, then x is in B:
- Answer: False.
- Example: Let A = {3, 5, 7} and B = {3, 4, 6}. Here, 5 is in A and A is not a subset of B, but 5 is not in B.
- If A is a subset of B and x is not in B, then x is not in A:
- Answer: True.
- Proof: If x were in A, then x would have to be in B because A is a subset of B. This contradicts the fact that x is not in B.
Question 3: Show that B = C
Let A, B, and C be sets such that A union B = A union C and A intersection B = A intersection C. Show that B = C.
Answer:
- Let x be an element of B.
- Then x is in (A union B), and since (A union B) = (A union C), x is also in (A union C).
- Case 1: If x is in A, then x is in (A intersection B). Since (A intersection B) = (A intersection C), then x is also in (A intersection C), so x is in C.
- Case 2: If x is not in A, since x is in (A union C), it must be in C.
- Therefore, B is a subset of C. By following the same logic, we can show C is a subset of B, proving B = C.
Question 8: Intersection and Equality
Show that A intersection B = A intersection C does not necessarily imply B = C.
Answer:
- Let A = {0, 1}, B = {0, 2, 3}, and C = {0, 4, 5}.
- Then (A intersection B) = {0} and (A intersection C) = {0}.
- Even though the intersections are equal, B is not equal to C because they have different elements (like 2 and 4).
Question 10: Pairwise Intersection
Find sets A, B, and C such that (A intersection B), (B intersection C), and (A intersection C) are non-empty, but (A intersection B intersection C) is empty.
Answer:
- Let A = {0, 2}, B = {1, 2}, and C = {2, 0}.
- (A intersection B) = {1} (non-empty).
- (B intersection C) = {1, 2} (non-empty).
- (A intersection C) = {0} (non-empty).
- However, there is no element common to all three sets, so (A intersection B intersection C) = empty set.
FAQs – Class 11 Maths Miscellaneous Exercise Chapter 1 - Sets
Q1. What is the focus of the Miscellaneous Exercise in Chapter 1?
The Miscellaneous Exercise in Chapter 1 focuses on reviewing and reinforcing the concepts related to sets, including set operations, types of sets, and applications of Venn diagrams.
Q2. What are the different types of sets covered in this chapter?
The chapter covers the following types of sets:
-
Finite and Infinite Sets
-
Subset and Power Set
-
Universal Set
-
Null Set (Empty Set)
-
Singleton Set
-
Equal Sets
-
Disjoint Sets
Q3. What operations can be performed on sets?
The main operations that can be performed on sets include:
-
Union (
A∪B): The set of all elements that belong to either set A or set B.
-
Intersection (
A∩B): The set of all elements that belong to both set A and set B.
-
Difference (
A−B): The set of all elements that belong to set A but not to set B.
-
Complement (
A′): The set of all elements that are in the universal set but not in set A.
Q4. Why is this exercise important for board exams?
This exercise is important because it provides practice in applying fundamental set theory concepts, which form the foundation for more complex topics in mathematics. Questions on sets are frequently asked in the Class 11 exams and are crucial for topics like probability and relations.
Q5. How can students prepare effectively for this Miscellaneous Exercise?
-
Review the definitions of different types of sets and their properties.
-
Practice set operations like union, intersection, and difference.
-
Solve problems involving Venn diagrams and set identities.
-
Work on identifying complementary sets and understanding universal sets.