Students often turn to OP Malhotra Class 11 Solutions Chapter 1 Sets Ex 1(c) to clarify doubts and improve problem-solving skills.
S Chand Class 11 ICSE Maths Solutions Chapter 1 Sets Ex 1(c)
Question 1.
Suggest a universal set for each of the following :
(i) {Jaipur, Chennai, Bangalore, Itanagar}
(ii) {Narmada, Cauvery, Mahanadi, Jhelum}
(iii) {Asia, Europe, Antarctica}
(iv) {Earth, Mars, Venus}
(v) {0, 5, 10, 15, 20, 25}
Solution:
(i) Since given set consist of elements which are state capitals of India.
∴ Universal set – {state capitals of India}
(ii) elements of given sets are rivers of India
∴ Universal set = {Rivers of India}
(iii) elements of given sets are continents
∴ Universal set can be taken as {continents}
(iv) each element of given set be a planet of our solar system
∴ Universal set can be taken as :
{planets of our solar system}
(v) given set containing some multiples of 5.
∴ Universal set = {x : x = 5p, p ∈ W, 0 ≤ p ≤ 10}
Question 2.
What universal set may be proposed for the following sets?
(i) The set of parallelograms
(ii) The set of irrational numbers
(iii) The set of positive even numbers
Solution:
(i) given set be a set of all parallelograms
∴ Universal set can be taken as set of all quadrilaterals.
since every parallelogram is a quadrilateral.
(ii) Since real numbers consists of rational and irrational numbers.
Hence universal set can be taken as set of all real numbers.
(iii) Since set of all natural numbers consist of all positive even numbers.
∴ Universal set can be taken as set of all natural numbers.
Question 3.
Solve the following equations :
(i) {x | 2x + 6 = 0, x ∈ Z}
(ii) {x | 5x + 16 = 1, x ∈ N}
(iii) {x | 2x – 3 < 7, x ∈ W}
(iv) {x | 4x – 25 > 13, x ∈ W}
(v) { y|\(\frac { 5y }{ 3 }\) – 7 ≤ 13, y is a prime number}
Solution:
(i) Given 2x + 6 = 0 and x ∈ Z
∴ 2x = – 6 ⇒ x = – 3
∴ Soln. of given set = {- 3}
(ii) Given x ∈ N, 5x + 16 = 1 ⇒ 5x = – 15 ⇒ x = – 3 ∉ N
∴ Solution set = Φ
(iii) Given 2x – 3 < 7 ⇒ 2x < 10 ⇒ x < 5, x ∈ W
∴ x = 0, 1, 2, 3, 4 Thus solution set = {0, 1, 2, 3, 4}
(iv) Given 4x – 25 > 13 ⇒ 4x > 38
⇒ x > \(\frac { 38 }{ 4 }\) = \(\frac { 19 }{ 2 }\) and x ∈ Z
∴ x = 10, 11, 12, ………
Thus solution set = {10, 11, 12, ….}
(v) Given \(\frac { 5y }{ 3 }\) – y ≤ 13
⇒ \(\frac { 5y }{ 3 }\) ≤ 20 ⇒ 5y ≤ 60
⇒ y < 20 and y is a prime number
∴ y = 2, 3, 5, 7, 11, 13, 17, 19
Thus solution set = {2, 3, 5, 7, 11, 13, 17, 19}
Question 4.
ξ = \(\left\{-2 \frac{1}{2},-1, \sqrt{2}, 3.5, \sqrt{30}, \sqrt{36}\right\}\)
X = {inters}, Y = {irrational numbers}
List the members of (i) X (ii) Y
Solution:
Given ξ = \(\left\{-2 \frac{1}{2},-1, \sqrt{2}, 3.5, \sqrt{30}, \sqrt{36}\right\}\)
(i) Given X = {integers} = {-1, \(\sqrt{36}\)} = {-1, 6}
(ii) Given Y = {irrational numbers} = \(\{\sqrt{2}, \sqrt{30}\}\)
Question 5.
ξ = {40, 41, 42, 43, 44, 45, 46, 47, 48, 49}
A = {prime numbers}, B = {odd numbers}
(i) Place the ten numbers in the correct places on the diagram.
(ii) Write the set B ∩ A.
Solution:
(i) Given ξ = {40, 41, 42, 43, 44, 45, 46, 47, 48, 49}
A = {prime numbers = {41, 43, 47}
and B = {odd numbers} = {41, 43, 45, 47, 49}
(ii) ∴ A’ = ξ – A = {40, 42, 44, 45, 46, 48, 49}
Thus B ∩ A’ = {45, 49}
Question 6.
Shade the regions as directed?
(i) A ∩ B
(ii) (A ∪ B)’
(iii) Complementary set B
Solution:
Question 7.
On the Venn diagrams shade the regions :
(i) A’ ∩ C’
(ii) (A ∪ C)∩B
Solution:
Question 8.
(i) On a copy of the Venn diagram, shade the set A∪(B ∩ C).
(ii) Express in set notation the subset shaded in the Venn diagram.
(iii) Express in set notation, as simply as possible, the subset shaded in the Venn diagram.
Solution:
(i)
(ii) shaded portion consist of P but not belonging to Q
∴ Required set = P – Q = P ∩ Q’
(iii) Required set = B ∩ C ∩ A’
Question 9.
Answer true or false. Refer to the given below figure.
(i) A ⊂ D
(ii) B ⊂ C
(iii) A ⊄ B
(iv) C ⊂ D
Solution:
(i) A ∩ D = A
∴ A ⊂ D
∴ given statement is true,
(ii) False, since every element of C is an element of B
∴ C ⊂ B
(iii) True, clearly every member of A is not a member of B,
(iv) True, Clearly from figure, every element of set C is a member of set D.
Question 10.
Describe the shaded areas in the following Venn diagrams.
Solution:
(i) shaded portion consist of elements of A but not belonging to set B
∴ required set = A ∩ B’
(ii) Clearly shaded portion consist of all elements which does not belong to set A, B and C i.e. their union.
∴ required set = (A∪B∪ C)’
(iii) shaded portion consists of common elements of A and B’ and C i.e. required equivalent set beA∩B’∩C1
(iv) Shaded portion consists of common elements of C’, A ⊂ B i. e. required set = C’ ∩ A ∩ B
Question 11.
Use the given diagram to shade the following regions.
(i) A’ ∩ B ∩ C
(ii) A’ ∩ B ∩ C’
(iii) (A ∩ B ∩ C)’
Solution:
Question 12.
X = {all triangles}, P = {isosceles triangles}, Z = {equilateral triangles}, Draw a Venn diagram to illustrate the relationship between these sets.
Solution:
Given X = {all triangles}
P = {isosceles triangles}
Z = {equilateral triangles}
Clearly P ⊂ X, Z ⊂ X, Also every equilateral triangle be an isosceles
∴ P ⊂ Z
Question 13.
Draw a Venn diagram to show the relationship between the following sets :
ξ = {quadrilaterals}, A = {parallelograms}, B = {rectangles}, C = rhombuses
Show in your diagram the region that represents the set of squares.
Solution:
Given ξ = {quadrilaterals}
A = {parallelograms}
B = {rectangles} set
C = {rhombuses}
Since every rectangle, rhombus, square in a plane is a parallelogram. Further every square is a rectangle.
Question 14.
Using ξ = {books}, N = {novels} and D = {detective novels}, represent the following statement as a Venn diagram, ‘Some novels are not detective novels.
Solution:
Given ξ = {books}
N = {Novels}
D = {detective novels}
shaded portion represents the non-detective novels.
Question 15.
Illustrate by a Venn diagram the rela- tionship between the sets A, B and C given that B ⊂ A, C ⊂ B’ and A ∩ C = Φ.
Solution:
Clearly B ⊂ A, A ∩ C = Φ, C ⊂ B’
Question 16.
If ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 3, 6, 10}, B = {1, 3, 5, 7, 9} and C = {1, 4, 9} then list the elements of the following sets.
(i) A ∩ B
(ii) B ∪ C
(iii) C’
(iv) n(B)
(v) B ∩ C
(vi) A ∩ C
(vii) A ∪ C
(ix) (A ∩ B)’
(x) A – B
(xi) B – A
(xii) (A ∪ B ∪ C)
(xiii) (A ∩ B ∩ C)’
(xiv) A ∆ B
Solution:
Given ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {1, 3, 6, 10} ;
B = {1, 3, 5,7, 9}
and C = {1, 4, 9}
(i) A ∩ B = {1, 3, 6, 10} ∩ {1, 3, 5, 7, 9}
= {1, 3}
(ii) B ∪ C = set of all distinct elements of B or C = {1, 3, 5, 7, 9} ∪ {1, 4, 9}
= {1, 3, 4, 5, 7, 9}
(iii) C’ = {x : x ∈ ξ, x ∉ C} = ξ – C
= {2, 3, 5, 6, 7, 8, 10}
(iv) n (B) = number of distinct elements in B
= 5
(v) B ∩ C = {x : x ∈ B and x ∈ C} = set of all common elements of B and C = {1, 9}
(vi) A ∩ C = set of all common elements of A and C = {1, 3, 6, 10} ∩ {1, 4, 9} = {1}
(vii) A ∪ C = set of all elements of A or C or both A and C = {x : x ∈ A or x ∈ C}
= {1, 3, 6, 10} ∪ {1, 4, 9}
= {1, 3, 4, 6, 9, 10}
(viii) A ∪ B ∪ C = {x : x ∈ A or x ∈ B or x ∈ C}
= {1, 3, 6, 10} ∪ {1, 3, 5, 7, 9} ∪ {1, 4, 9}
= {1, 3, 4, 5, 6, 7, 9, 10}
(ix) (A ∩ B)’ = ξ – (A ∩ B) = set of all elements of set ξ but not in A ∩ B
= ξ – {1, 3} = {2, 4, 5, 6, 7, 8, 9, 10}
(x) A – B = set of all elements which are in A but not in B
= {1, 3, 6, 10} – {1, 3, 5, 7, 9}
= {6, 10}
(xi) B – A = set of all elements which are in B but not in A
= {1, 3, 5, 7, 9} – {1, 3, 6, 10}
= {5, 7, 9}
(xii) A∪B∪C = {1, 3, 6, 10} ∪ {1, 3, 5, 7, 9} ∪ {1, 4, 9}
= {1, 3, 4, 5, 6, 7, 9, 10}
∴ (A∪B∪C)’ = ξ – (A ∪ B ∪ C)
= {2, 8}
(xiii) A∩B∩C= {1,3, 6, 10} ∩ {1, 3, 5, 7, 9} ∩ {1, 4, 9}
= {1}
∴ (A ∩ B ∩ C)’ = ξ – (A ∩ B ∩ C) = set of all elements which are in ξ but not in A∩B∩C = {2, 3 4, 5, 6, 7, 8, 9, 10}
(xiv) A ∆ B = (A – B) ∪ (B – A)
= {6, 10} ∪ {5, 7, 9} = {5, 6, 7, 9,10} [using (x) and (xi)]
Question 17.
Let L = {Setters of CRICKET}, M = {Setters of CATERPILLAR} and N = {Setters of CARE TAKER} find : (i) L∪M (ii) M ∪ N (iii) L∪N
Solution:
Given L = {C, R, I, K, E, T}
M = {C, A, T, E, R, P, I, L} and N = {C, A, R, E, T, K}
(i) L∪M = {C,R, I, K, E, T} ∪ {C, A, T, E, R, P, I, L}
= {C, A, T, E, I, K, R, P, L}
(ii) M∪N = {C, A, T, E, R, P, I, L} ∪ {C, A, R, E, T, K}
= {C, A, T, E, R, P, I, L, K}
(iii) L∪N = {C,R, I, K, E, T} ∪ {C, A, R, E, T, K} = {C, R, I, K, E, T, A}
Question 18.
Taking the set of natural numbers as the universal set, write down the complements of the following sets :
(i) {x : x is a positive multiple of 3}
(ii) {x : x is a prime number}
(iii) {x : x is a natural number divisible by 3 and 5}
(iv) {x : x + 5 – 8}
(v) {x : 2x + 5 = 9}
(vi) {x : x ≥ 7}
(vii) {x : x ∈ N and 2x + 1 > 10}
Solution:
Universal set = N = set of natural numbers
(i) Let A = {x: x is a positive multiple of 3}
∴ A’ = {x: x is not a positive multiple of 3 }
(ii) Let B = {x : x is a prime number}
∴ B’ = {x ; x is not a prime number}
(iii) Let C = {x: x is a natural number which is divisible by 3 and 5}
∴ C’ = {x : x is a natural number which is not divisible by 15}
(iv) Let D = {x : x + 5 = 8}
Since x + 5 = 8 ⇒ x = 3
∴ D’ = {x : x ∈ N and x ≠ 3}
(v) Let E = {x : 2x + 5 = 9}
since 2x + 5 = 9 ⇒ 2x = 4 ⇒ x = 2
∴ E’ = {x : x ∈ N and x ≠ 2}
(vi) Let F = {x : x ≥ 7}
∴ F’ = {x : x ∈ N and x < 7}
= {1, 2, 3, 4, 5, 6}
(vii) Let G = {x : x ∈ N and 2x + 1 > 10}
since 2x + 1 > 10 ⇒ 2x > 9
⇒ x > \(\frac { 9 }{ 2 }\) and x ∈ N
∴ x = 5, 6, 7, 8, ….
∴ G’ = {x : x ∈ N and 2x + 1 ≤ 10} = {1, 2, 3, 4}
Question 19.
Refer to the Venn diagram. List the elements of the following sets.
(i) (A ∩ B) ∪ C
(ii) A ∩ (B ∪ C)
(iii) (A ∩ C) ∩ (B ∩ C)
(iv) (A ∪ B) ∩ (B ∩ C)
(v) B – C
(vi) C – B
(vii) B ∆ C
Solution:
From Venn diagram
A = {-1, -2, -3, -4, 3, 4} ; B = {0, – 1, – 2, 3} and C = {3, 4, 5, 6}
(i) A ∩ B = {-2, – 1, 3}
∴ (A∩B)∪C = {-2, -1, 3, 4, 5, 6}
(ii) B∪C = {-1, -2, 0, 3, 4, 5, 6}
∴ A ∩ (B ∪ C) = {- 1, – 2, – 3, – 4, 3, 4} ∩ {-1,-2, 0,3,4, 5, 6} = {-1,-2, 3,4}
(iii) A ∩ C = {- 1, – 2, – 3, – 4, 3, 4} ∩ {3, 4, 5, 6} = {3, 4}
B∩C = {0,- 1,-2, 3} ∩ {3,4, 5,6} = {3}
∴ (A∩C)∩(B∩C)= {3,4} ∩ {3} = {3}
(iv) A∪B = {-1, -2, -3, -4, 3, 4} ∪ {0, -1, -2, 3}
= {0, -1, -2, -3, -4, 3, 4}
B ∩ C = (0, – 1, – 2, 3} ∩ {3, 4, 5, 6} = {3}
∴ (A ∪B) ∩ (B∩C) = {3}
(v) B – C = {0, – 1, – 2, 3} – {3, 4, 5,6}
= {0, -1, -2}
(vi) C – B = {3, 4, 5, 6} – {0, – 1, – 2, 3}
= {4, 5, 6}
(vii) B ∆ C = (B – C) ∪ (C – B)
= {0, – 1, – 2} ∪ {4, 5, 6}
= {- 2, – 1, 0, 4, 5, 6}
Question 20.
If ξ = (1, 2, 3, 4}, A = {1, 4}, B = {1, 3}, then list the elements of
(i) A’
(ii) B’
(iii) (A ∩ B)’
(iv) (A ∪ B)’
(v) A’ ∩ B’
(vi) A’ ∪ B’
Also show that
(A ∩ B)’ = A’ ∪ B
and (A ∪ B)’ = A’ ∩ B
Solution:
Given ξ = {1, 2, 3, 4} ; A = {1, 4} and B = {1, 3}
(i) A’ = ξ – A = {1, 2, 3, 4} – {1, 3} = {2, 3}
(ii) B’ = ξ – B = {1, 2, 3, 4} – {1, 3} = (2, 4}
(iii) A∩B = {1, 4} ∩ {1, 3} = {1}
(A ∩ B)’ = ξ – (A ∩ B) = {2, 3, 4}
(iv) A∪B = {1, 4} ∪ {1, 3} = {1, 3, 4}
∴ (A∪B)’ = ξ – (A∪B)
= {1,2, 3, 4} – {1,3, 4} = {2}
(v) A’∩B’= {2, 3} ∩ {2, 4} = {2}
[using part (i) and (if)]
(vi) A’ ∪ B’ = {2, 3} ∪ {2, 4} = {2, 3, 4}
using part (iii) and part (vi) ; we have
(A ∩ B)’ = A’ ∪ B’