NCERT Solutions for Class 11 Maths Ch-1 Sets
NCERT Book Important Questions for Class 11 Maths Chapter 1 Sets
Thank you for reading this post, don't forget to subscribe!Exercise 1.1
Question 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.
(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
Answer
(i) The collection consists of months January, June and July. It is well-defined and therefore, it is a set.
(ii) Ten most talented writers of India vary from person to person. It is not well-defined and therefore, it is not a set.
(iii) A team of eleven best-cricket batsmen of the world vary from person to person. It is not well-defined and therefore, it is not a set.
(iv) The given collection is well-defined and therefore, it is a set.
(v) The collection consists of first 99 natural numbers. It is well-defined and therefore, it is a set.
(vi) It is a well-defined and therefore, it is a set.
(vii) It is a well-defined and therefore, it is a set.
(viii) It is a well-defined and therefore, it is a set.
(ix) The most dangerous animals of the world vary from person to person. It is not well-defined and therefore, it is not a set.
NCERT Solutions for Class 11 Maths Ch-1 Sets
Question 2:
Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol \(\in\) or \(\notin\) in the blank spaces:
(i) 5. . .A (ii) 8 . . . A (iii) 0. . .A (iv) 4. . . A (v) 2. . .A (vi) 10. . .A
Answer
(i) 5 \(\in \) A (ii) 8 \(\notin \) A (iii) 0 \(\notin \) A (iv) 4 \(\in\) A (v) 2 \(\in\) A (vi) 10 \(\notin\) A
Question 3:
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
Answer
(i) A = {x : x is an integer and –3 ≤ x < 7}
\(\quad\) = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}
(ii) B = {x : x is a natural number less than 6}
\(\quad\) = {1, 2, 3, 4, 5}
(iii) C = {x : x is a two-digit natural number such that the sum of its digits is 8}
\(\quad\) = {17, 26, 35, 44, 53, 62, 71, 80}
(iv) D = {x : x is a prime number which is divisor of 60}
\(\quad\) = {2, 3, 5}
(v) E = The set of all letters in the word TRIGONOMETRY
\(\quad\) = {T, R, I, G, O, N, M, E, Y}
(vi) F = The set of all letters in the word BETTER
\(\quad\) = {B, E, T, R}
Question 4:
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}
Answer
(i) (3, 6, 9, 12} = {x : x = 3n, n \(\in \) N and \( 1 \leq n \leq 4 \)}
(ii) {2,4,8,16,32} = {21, 22, 23, 24, 25} = {x : x = 2n, n \(\in \) N and \( 1 \leq n \leq 5 \)}
(iii) {5, 25, 125, 625} = {51, 52, 53, 54} = {x : x = 5n, n \(\in \) N and \( 1 \leq n \leq 4 \)}
(iv) {2, 4, 6, . . .} = {x : x = 2n, n \(\in \) N}
(v) {1,4,9, . . .,100} = {11, 22, 32, … , 102} = {x : x = n2, n \(\in \) N and \( 1 \leq n \leq 10 \)}
Question 5:
List all the elements of the following sets :
(i) A = {x : x is an odd natural number}
(ii) B = {x : x is an integer, -1/2< x <9/2}
(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 }.
Answer
(i) A = {x : x is an odd natural number} = {1, 3, 5, …}
(ii) B = {x : x is an integer, -1/2< x <9/2} = {1, 1, 2, 3, 4}
(iii) C = {x : x is an integer, x2 ≤ 4} = {-2, -1, , 1, 2}
(iv) D = {x : x is a letter in the word “LOYAL”} = {L, O, Y, A}
(v) E = {x : x is a month of a year not having 31 days} = {February, April, June, September, November}
(vi) F = {x : x is a consonant in the English alphabet which precedes k } = {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 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)-(c); (ii)-(a); (iii)-(d); (iv)-(b)
NCERT Solutions for Class 11 Maths Ch-1 Sets
Exercise 1.2
Question 1:
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}
Answer
(i) There is no odd natural number which is divisible by 2. So, the given set is a null set.
(ii) Set of even prime numbers = {2}.
So, the given set is not a null set. It is a singleton set.
(iii) There is no natural number which is both less than 5 and greater than 7.
So, the given set is a null set.
(iv) Two Parallel have no common point.
So, the given set is a null set.
Question 2:
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
Answer
(i) Since there are 12 months in a year, the given set is finite.
(ii) Since the number of elements in the set is infinite, the given set is infinite.
(iii) Since the number of elements in the set is 10, the given set is finite.
(iv) Since there are infinitely many numbers greater than 100, the given set is infinite.
(v) Since the number of prime less than 99 is a definite number, the given set is finite.
Question 3:
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)
Answer
(i) Since there are infinite number of lines parallel to hte x-axis, the given set is infinite.
(i) Since there are 26 letters i.e., definite number of letters in the English alphabet, the given set is finite.
(i) Since there are infinitely many multiple of 5, the given set is infinite.
(i) Since there are infinitely many animals living on the earth, the given set is infinite.
(i) There is no end to the number of circles passing the origin (0, 0). Hence, the given set is infinite.
Question 4:
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, . . . }
Answer
(i) A = B
(ii) A \(\neq \) B
(iii) A = B
(iv) A \(\neq \) B
Question 5:
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}
Answer
(i) A = {2, 3}
and B {-2, -3}
A \(\neq \) B
(ii) A = {F, O, L, W} and B = {W, O, L, F}
A = B
Question 6:
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}
Answer
B = D; E = G
NCERT Solutions for Class 11 Maths Ch-1 Sets
Exercise 1.3
Question 1:
Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces :
(i) { 2, 3, 4 } . . . { 1, 2, 3, 4,5 }
(ii) { a, b, c } . . . { b, c, d }
(iii) {x : x is a student of Class XI of your school}. . .{x : x student of your school}
(iv) {x : x is a circle in the plane} . . .{x : x is a circle in the same plane with radius 1 unit}
(v) {x : x is a triangle in a plane} . . . {x : x is a rectangle in the plane}
(vi) {x : x is an equilateral triangle in a plane} . . . {x : x is a triangle in the same plane}
(vii) {x : x is an even natural number} . . . {x : x is an integer}
Answer
(i) ⊂ (ii) ⊄ (iii) ⊂ (iv) ⊄ (v) ⊄ (vi) ⊂ (vii) ⊂
NCERT Solutions for Class 11 Maths Ch-1
Question 2:
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}
Answer
(i) False (ii) True (iii) False (iv) True (v) False (vi) True
Question 3:
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
Answer
A = { 1, 2, { 3, 4 }, 5 }
(i) {3, 4} ⊂ A is incorrect
Since 3 \(\notin\) A and 4 \(\notin\) A,
\(\therefore\) A ⊄ B
(ii) {3, 4} ∈A is correct
{3, 4} is an element of A.
(iii) {{3, 4}} ⊂ A is correct
{3, 4} ∈ {{3, 4}} and {3, 4} ∈ A.
(iv) 1∈A is correct
1 is an element of A.
(v) 1⊂ A is incorrect
An element of a set can never be a subset of itself.
(vi) {1, 2, 5} ⊂ A is correct
Each element of {1, 2, 5} is also an element of A.
(vii) {1, 2, 5} ∈ A is incorrect
{1, 2, 5} is not an element of A.
(viii) {1, 2, 3} ⊂ A is incorrect
3 ∈ {1, 2, 3}; where, 3 ∉ A.
(ix) Φ ∈ A is incorrect
Φ is not an element of A.
(x) Φ ⊂ A is correct
Φ is a subset of every set.
(xi) {Φ} ⊂ A is incorrect
Φ∈ {Φ}; where, Φ ∈ A.
Question 4:
Write down all the subsets of the following sets
(i) {a} (ii) {a, b} (iii) {1, 2, 3} (iv) φ
Answer
(i) Subsets of {a} are Φ and {a}.
(ii) Subsets of {a, b} are Φ, {a}, {b}, and {a, b}.
(iii) Subsets of {1, 2, 3} are Φ, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, and {1, 2, 3}.
(iv) Only subset of Φ is Φ.
Question 5:
Write the following as intervals :
(i) {x : x ∈ R, – 4 < x ≤ 6} \(\quad\) (ii) {x : x ∈ R, – 12 < x < –10}
(iii) {x : x ∈ R, 0 ≤ x < 7} \(\quad\) (iv) {x : x ∈ R, 3 ≤ x ≤ 4}
Answer
(i) {x: x ∈ R, –4 < x ≤ 6} = (–4, 6]
(ii) {x: x ∈ R, –12 < x < –10} = (–12, –10)
(iii) {x: x ∈ R, 0 ≤ x < 7} = [0, 7)
(iv) {x: x ∈ R, 3 ≤ x ≤ 4} = [3, 4]
Question 6:
Write the following intervals in set-builder form :
(i) (– 3, 0) (ii) [6, 12] (iii) (6, 12] (iv) [–23, 5)
Answer
(i) (–3, 0) = {x: x ∈ R, –3 < x < 0}
(ii) [6, 12] = {x: x ∈ R, 6 ≤ x ≤ 12}
(iii) (6, 12] ={x: x ∈ R, 6 < x ≤ 12}
(iv) [–23, 5) = {x: x ∈ R, –23 ≤ x < 5}
Question 7:
What universal set(s) would you propose for each of the following :
(i) The set of right triangles. (ii) The set of isosceles triangles.
Answer
The set of all triangles
Question 8:
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}
Answer
(i) {0, 1, 2, 3, 4, 5, 6}
A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}
A ⊂ {0, 1, 2, 3, 4, 5, 6}
and B ⊂ {0, 1, 2, 3, 4, 5, 6}
So C ⊄ {0, 1, 2, 3, 4, 5, 6}
Hence, the set {0, 1, 2, 3, 4, 5, 6} cannot be the universal set for the sets A, B, and C.
(ii) A ⊄ Φ, B ⊄ Φ, C ⊄ Φ
Hence, Φ cannot be the universal set for the sets A, B, and C.
(iii) A ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
B ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
C ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Hence, the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is the universal set for the sets A, B, and C.
(iv) A ⊂ {1, 2, 3, 4, 5, 6, 7, 8}
B ⊂ {1, 2, 3, 4, 5, 6, 7, 8}
So C ⊄ {1, 2, 3, 4, 5, 6, 7, 8}
Hence, the set {1, 2, 3, 4, 5, 6, 7, 8} cannot be the universal set for the sets A, B, and C.
NCERT Solutions of Class 11 Maths Ch-1 Sets
Exercise 1.4
uestion 1:
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}
\(\quad\) B = {x : x is a natural number less than 6}
(iv) A = {x : x is a natural number and 1 < x ≤6 }
\(\quad\) B = {x : x is a natural number and 6 < x < 10 }
(v) A = {1, 2, 3}, B = φ
Answer
(i) X = {1, 3, 5} Y = {1, 2, 3}
\(\Rightarrow\) X ∪ Y= {1, 2, 3, 5}
(ii) A = {a, e, i, o, u} B = {a, b, c}
\(\Rightarrow\) A∪ B = {a, b, c, e, i, o, u}
(iii) A = {x: x is a natural number and multiple of 3} = {3, 6, 9 …}
\(\quad\) B = {x: x is a natural number less than 6} = {1, 2, 3, 4, 5, 6}
\(\Rightarrow\) A ∪ B = {1, 2, 4, 5, 3, 6, 9, 12 …}
(iv) A = {x: x is a natural number and 1 < x ≤ 6} = {2, 3, 4, 5, 6}
\(\quad\)B = {x: x is a natural number and 6 < x < 10} = {7, 8, 9}
\(\Rightarrow\) A∪ B = {2, 3, 4, 5, 6, 7, 8, 9}
(v) A = {1, 2, 3}, B = Φ
\(\Rightarrow\) A∪ B = {1, 2, 3}
Question 2:
Let A = { a, b }, B = {a, b, c}. Is A ⊂ B ? What is A ∪ B ?
Answer
A = {a, b} and B = {a, b, c}
Yes, A ⊂ B
A∪ B = {a, b, c} = B
Question 3:
If A and B are two sets such that A ⊂ B, then what is A ∪ B ?
Answer
If A and B are two sets such that A ⊂ B, then A ∪ B = B.
Question 4:
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
Answer
A = {1, 2, 3, 4], B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}
(i) A ∪ B = {1, 2, 3, 4, 5, 6}
(ii) A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}
(iii) B ∪ C = {3, 4, 5, 6, 7, 8}
(iv) B ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}
(v) A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}
(vi) A ∪ B ∪ D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(vii) B ∪ C ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}
Question 5:
Find the intersection of each pair of sets of question 1 above.
Answer
(i) X = {1, 3, 5}, Y = {1, 2, 3}
\(\Rightarrow\) X ∩ Y = {1, 3}
(ii) A = {a, e, i, o, u}, B = {a, b, c}
\(\Rightarrow\) A ∩ B = {a}
(iii) A = {x: x is a natural number and multiple of 3} = (3, 6, 9 …}
\(\quad\) B = {x: x is a natural number less than 6} = {1, 2, 3, 4, 5}
\(\Rightarrow\) A ∩ B = {3}
(iv) A = {x: x is a natural number and 1 < x ≤ 6} = {2, 3, 4, 5, 6}
\(\quad\) B = {x: x is a natural number and 6 < x < 10} = {7, 8, 9}
\(\Rightarrow\) A ∩ B = Φ
(v) A = {1, 2, 3}, B = Φ
\(\Rightarrow\) A ∩ B = Φ
Question 6:
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)
Answer
(i) A ∩ B = {7, 9, 11}
(ii) B ∩ C = {11, 13}
(iii) A ∩ C ∩ D = {A ∩ C} ∩ D = {11} ∩ {15, 17} = Φ
(iv) A ∩ C = {11}
(v) B ∩ D = Φ
(vi) A ∩ (B ∪ C)
\(\quad\) (B ∪ C) = {7, 9, 11, 13, 15}
\(\quad\) A ∩ (B ∪ C) = {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15} = {7, 9, 11}
(vii) A ∩ D = Φ
(viii) A ∩ (B ∪ D)
\(\quad\) (B ∪ D) = {7, 9, 11, 13, 15, 17}
\(\quad\) A ∩ (B ∪ D) = {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15, 17} = {7, 9, 11}
(ix) (A ∩ B) ∩ (B ∪ C) = {7, 9, 11} ∩ {7, 9, 11, 13, 15} = {7, 9, 11}
(x) (A ∪ D) ∩ (B ∪ C) = {3, 5, 7, 9, 11, 15, 17) ∩ {7, 9, 11, 13, 15} = {7, 9, 11, 15}
Question 7:
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
Answer
A = {x: x is a natural number} = {1, 2, 3, 4, 5 …}
B ={x: x is an even natural number} = {2, 4, 6, 8 …}
C = {x: x is an odd natural number} = {1, 3, 5, 7, 9 …}
D = {x: x is a prime number} = {2, 3, 5, 7 …}
(i) A ∩B = {x: x is a even natural number} = B
(ii) A ∩ C = {x: x is an odd natural number} = C
(iii) A ∩ D = {x: x is a prime number} = D
(iv) B ∩ C = Φ
(v) B ∩ D = {2}
(vi) C ∩ D = {x: x is odd prime number}
Question 8:
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}
Answer
(i) {1, 2, 3, 4} and {x : x is a natural number and 4 ≤ x ≤ 6} = {4, 5, 6}
\(\quad\) {1, 2, 3, 4} ∩ {4, 5, 6} = {4}
Hence, this pair of sets is not disjoint.
(ii) {a, e, i, o, u} ∩ (c, d, e, f} = {e}
Hence, {a, e, i, o, u} and (c, d, e, f} are not disjoint.
(iii) {x: x is an even integer} ∩ {x: x is an odd integer} = Φ
Hence, this pair of sets is disjoint.
Question 9:
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
Answer
(i) A – B = {3, 6, 9, 15, 18, 21}
(ii) A – C = {3, 9, 15, 18, 21}
(iii) A – D = {3, 6, 9, 12, 18, 21}
(iv) B – A = {4, 8, 16, 20}
(v) C – A = {2, 4, 8, 10, 14, 16}
(vi) D – A = {5, 10, 20}
(vii) B – C = {20}
(viii) B – D = {4, 8, 12, 16}
(ix) C – B = {2, 6, 10, 14}
(x) D – B = {5, 10, 15}
(xi) C – D = {2, 4, 6, 8, 12, 14, 16}
(xii) D – C = {5, 15, 20}
Question 10:
If X= { a, b, c, d } and Y = { f, b, d, g}, find (i) X – Y (ii) Y – X (iii) X ∩ Y
Answer
(i) X – Y = {a, c}
(ii) Y – X = {f, g}
(iii) X ∩ Y = {b, d}
Question 11:
If R is the set of real numbers and Q is the set of rational numbers, then what is R – Q?
Answer
R – Q is a set of irrational numbers.
Question 12:
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.
Answer
(i) False
If 3 ∈ {2, 3, 4, 5}, 3 ∈ {3, 6}
So we get {2, 3, 4, 5} ∩ {3, 6} = {3}
(ii) False
If a ∈ {a, e, i, o, u}, a ∈ {a, b, c, d}
So we get {a, e, i, o, u} ∩ {a, b, c, d} = {a}
(iii) True
Here {2, 6, 10, 14} ∩ {3, 7, 11, 15} = Φ
(iv) True
Here {2, 6, 10} ∩ {3, 7, 11} = Φ
NCERT Solutions for Class 11 Maths Ch-1 Sets
Exercise 1.5
Question 1:
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)′
Answer
We are given that
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}
(i) A’ = {5, 6, 7, 8, 9}
(ii) B’ = {1, 3, 5, 7, 9}
(iii) A U C = {1, 2, 3, 4, 5, 6}
So we get (A U C)’ = {7, 8, 9}
(iv) A U B = {1, 2, 3, 4, 6, 8}
So we get, (A U B)’ = {5, 7, 9}
(v) (A’)’ = A = {1, 2, 3, 4}
(vi) B – C = {2, 8}
So we get, (B – C)’ = {1, 3, 4, 5, 6, 7, 9}
Question 2:
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}
Answer
(i) A = {a, b, c}
\(\quad\) A’ = {d, e, f, g, h}
(ii) B = {d, e, f, g}
\(\quad\) B’ = {a, b, c, h}
(iii) C = {a, c, e, g}
\(\quad\) C’ = {b, d, f, h}
(iv) D = {f, g, h, a}
\(\quad\) D’ = {b, c, d, e}
Question 3:
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 }
Answer
U = N: Set of natural numbers
(i) {x: x is an even natural number}´ = {x: x is an odd natural number}
(ii) {x: x is an odd natural number}´ = {x: x is an even natural number}
(iii) {x: x is a positive multiple of 3}´ = {x: x ∈ N and x is not a multiple of 3}
(iv) {x: x is a prime number}´ ={x: x is a positive composite number and x = 1}
(v) {x: x is a natural number divisible by 3 and 5}´ = {x: x is a natural number that is not divisible by 3 or 5}
(vi) {x: x is a perfect square}´ = {x: x ∈ N and x is not a perfect square}
(vii) {x: x is a perfect cube}´ = {x: x ∈ N and x is not a perfect cube}
(viii) {x: x + 5 = 8}´ = {x: x ∈ N and x ≠ 3}
(ix) {x: 2x + 5 = 9}´ = {x: x ∈ N and x ≠ 2}
(x) {x: x ≥ 7}´ = {x: x ∈ N and x < 7}
(xi) {x: x ∈ N and 2x + 1 > 10}´ = {x: x ∈ N and x ≤ 9/2} = {1, 2, 3, 4}
Question 4:
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′
Answer
(i) (A U B)’ = {2, 3, 4, 5, 6, 7, 8}’ = {1, 9}
A’ ∩ B’ = {1, 3, 5, 7, 9} ∩ {1, 4, 6, 8, 9} = {1, 9}
Therefore, (A U B)’ = A’ ∩ B’.
(ii) (A ∩ B)’ = {2}’ = {1, 3, 4, 5, 6, 7, 8, 9}
A’ U B’ = {1, 3, 5, 7, 9} U {1, 4, 6, 8, 9} = {1, 3, 4, 5, 6, 7, 8, 9}
Therefore, (A ∩ B)’ = A’ U B’.
Question 5:
Draw appropriate Venn diagram for each of the following :
(i) (A ∪ B)′, (ii) A′ ∩ B′, (iii) (A ∩ B)′, (iv) A′ ∪ B′
Answer
Question 6:
Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is A′?
Answer
A’ is the set of all equilateral triangles.
Question 7:
Fill in the blanks to make each of the following a true statement :
(i) A ∪ A′ = . . . (ii) φ′ ∩ A = . . . (iii) A ∩ A′ = . . . (iv) U′ ∩ A = . . .
Answer
(i) A U A’ = U
(ii) Φ′ ∩ A = U ∩ A = A
(iii) A ∩ A’ = Φ
(iv) U’ ∩ A = Φ ∩ A = Φ
NCERT Solutions for Class 11 Maths Ch-1 Sets
Miscellaneous Exercise on Chapter 1
Question 1:
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 }.
Answer
A = {x: x ∈ R and x satisfies x2 – 8x + 12 =0} = {2, 6} , B = {2, 4, 6}, C = {2, 4, 6, 8 …}, D = {6}
Hence, D ⊂ A ⊂ B ⊂ C
Hence, A ⊂ B, A ⊂ C, B ⊂ C, D ⊂ A, D ⊂ B, D ⊂ C
Question 2:
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.
(i) If x ∈ A and A ∈ B , then x ∈ B
(ii) If A ⊂ B and B ∈ C , then A ∈ C
(iii) If A ⊂ B and B ⊂ C , then A ⊂ C
(iv) If A ⊄ B and B ⊄ C , then A ⊄ C
(v) If x ∈ A and A ⊄ B , then x ∈ B
(vi) If A ⊂ B and x ∉ B , then x ∉ A
Answer
(i) False
A = {1, 2} and B = {1, {1, 2}, {3}}
2 ∈ {1, 2} and {1, 2} ∈ {1, {1, 2}, {3}}
Hence, we get, A ∈ B
We also know, {2} ∉ {1, {1, 2}, {3}}
(ii) False
ATQ,
Let us assume that, A {2} B = {0, 2} and, C = {1, {0, 2}, 3}
From the question,
A ⊂ B
Hence, B ∈ C
But, we know, A ∉ C
(iii) True
ATQ,
A ⊂ B and B ⊂ C
Let us assume that, x ∈ A
Then, we have, x ∈ B and, x ∈ C
Therefore, A ⊂ C
(iv) False
ATQ,
A ⊄ B
Also, B ⊄ C
Let us assume that, A = {1, 2}, B = {0, 6, 8} and, C = {0, 1, 2, 6, 9}
∴ A ⊂ C
(v) False
ATQ,
x ∈ A
Also, A ⊄ B
Let us assume that, A = {3, 5, 7}
Also, B = {3, 4, 6}
We know that, A ⊄ B
∴ 5 ∉ B
(vi) True
ATQ,
A ⊂ B Also, x ∉ B
Let us assume that, x ∈ A,
We have, x ∈ B,
From the question,
We have, x ∉ B
∴ x ∉ A
Question 3:
Let A, B, and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. Show that B = C.
Question 4:
Show that the following four conditions are equivalent :
(i) A ⊂ B(ii) A – B = φ (iii) A ∪ B = B (iv) A ∩ B = A
Question 5:
Show that if A ⊂ B, then C – B ⊂ C – A.
Question 6:
Show that for any sets A and B, A = ( A ∩ B ) ∪ ( A – B ) and A ∪ ( B – A ) = ( A ∪ B )
Question 7:
Using properties of sets, show that (i) A ∪ ( A ∩ B ) = A (ii) A ∩ ( A ∪ B ) = A.
Question 8:
Show that A ∩ B = A ∩ C need not imply B = C.
Question 9:
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.
Question 10:
Find sets A, B and C such that A ∩ B, B ∩ C and A ∩ C are non-empty sets and A ∩ B ∩ C = φ.
NCERT Solutions for Class 11 Maths Ch-1 Sets
Comments are closed.