Problem Set 1
Question 1.Choose the correct alternative answer for each of the following questions.
If M = {1, 3, 5}, N = {2, 4, 6}, then M ∩ N =?
A. {1, 2, 3, 4, 5, 6}
B. {1, 3, 5}
C. ϕ
D. {2, 4, 6}
Answer:
Given: M = {1, 3, 5}, N = {2, 4, 6}
(M ∩ N) = set of all common elements of M and N
Hence, (M ∩ N) = ϕ
Option (C) is correct.
Question 2.
Choose the correct alternative answer for each of the following questions.
P = {x | x is an odd natural number,
How to write this set-in roster form?
A. {1, 3, 5}
B. {1, 2, 3, 4, 5}
C. {1, 3}
D. {3, 5}
Answer:
Given : P = {x | x is an odd natural number, 
set in roster form:
⇒ P = {3, 5}
Option (D) is correct.
Question 3.
Choose the correct alternative answer for each of the following questions.
P = {1, 2, ........., 10}, What type of set P is ?
A. Null set
B. Infinite set
C. Finite set
D. None of these
Answer:
Given : P = {1, 2, ........., 10}
As the number of elements given are finite.
⇒ given set is finite set.
Option (C) is correct.
Question 4.
Choose the correct alternative answer for each of the following questions.
M ∪ N= {1, 2, 3, 4, 5, 6} and M = {1, 2, 4} then which of the following represent set N?
A. {1, 2, 3}
B. {3, 4, 5, 6}
C. {2, 5, 6}
D. {4, 5, 6}
Answer:
Given : (M ∪ N) = {1, 2, 3, 4, 5, 6}, M = {1, 2, 4}
(M ∪ N) = set of all elements of set M and N
Hence, N = {3, 4, 5, 6}
Option (B) is correct.
Question 5.
Choose the correct alternative answer for each of the following questions.
If 
then Which of the following set represent P ∩(P ∪ M)?
A. P
B. M
C. P ∪ M
D. P ∩ M
Answer:
Given :
As,
Then using property,
(P ∪ M) = M
As,
Then using property,
(P ∩ M) = P
Question 6.
Choose the correct alternative answer for each of the following questions.
Which of the following sets are empty sets?
A. set of intersecting points of parallel lines
B. set of even prime numbers.
C. Month of an english calendar having less than 30 days.
D. 
Answer:
A. set of intersecting points of parallel lines = ϕ
Because no two parallel lines will intersect.
Hence, Option (A) is correct.
Question 7.
Find the correct option for the given question.
Which of the following collections is a set?
A. Colours of the rainbow
B. Tall trees in the school campus.
C. Rich people in the village
D. Easy examples in the book
Answer:
SET: If we can definitely and clearly decide the objects of a given collection
then that collection is called set.
Hence, option A is correct answer.
Because we can clearly say that colours of rainbow are seven in number as
(violet, indigo, blue, green, yellow, orange and red).
Hence, Option (A) is correct.
Question 8.
Find the correct option for the given question.
Which of the following set represent N ∩ W?
A. {1, 2, 3, .....}
B. {0, 1, 2, 3, ....}
C. {0}
(D) { }
Answer:
Given : N = {1, 2, .........∞} and W = {0, 1, 2, .........∞}
(N ∩ W) = set of all common elements of N and W
Hence, (N ∩ W) = {1, 2, .........∞}
Option (A) is correct.
Question 9.
Find the correct option for the given question.
P = {x | x is a letter of the word ' Indian'} then which one of the following is set P in listing form?
A. {i, n, d}
B. {i, n, d, a}
C. {i,n,d,i,a}
D. {n, d, a}
Answer:
Given : P = {x | x is a letter of the word ' indian'}
listing form = each of the element is written only once and separated by commas in curly brackets.
Hence, P = {i, n, d, a}
Option (B) is correct.
Question 10.
Find the correct option for the given question.
If T = {1, 2, 3, 4, 5} and M = {3, 4, 7, 8} then T ∪ M=?
A. {1, 2, 3, 4, 5, 7}
B. {1, 2, 3, 7, 8}
C. {1, 2, 3, 4, 5, 7, 8}
D. {3, 4}
Answer:
Given : T = {1, 2, 3, 4, 5} and M = {3, 4, 7, 8}
(T ∪ M) = set of all elements of set T and M
Hence, (T ∪ M) = {1, 2, 3, 4, 5, 7, 8}
Option (C) is correct.
Question 11.
Out of 100 persons in a group, 72 persons speak English and 43 persons speak French. Each one out of 100 persons speak at least one language. Then how many speak only English? How many speak only French? How many of them speak English and French both?
Answer:
Given: n(A) = 72, n(B) = 43 and n (A ∪ B) = 100
Total number of persons who speak at least one language = n (A ∪ B) = 100
n(A) = number of students who speaks English
n(B) = number of students who speaks French
n (A ∩ B) = number of students who speak both
As we know, n (A ∪ B) = n(A) + n(B) - n (A ∩ B)
⇒ n (A ∩ B) = 72 + 43 - 100
⇒ n (A ∩ B) = 115 - 100
⇒ n (A ∩ B) = 15
number of students who speak both the language = 15
number of students who speaks English only = 72 – 15 = 57
number of students who speaks French only = 43 – 15 = 28
Question 12.
70 trees were planted by Parth and 90 trees were planted by Pradnya on the occasion ofTree Plantation Week. Out of these; 25 trees were planted by both of them together. Howmany trees were planted by Parth or Pradnya?
Answer:
Given: n(A) = 70, n(B) = 90 and n (A ∩ B) = x = 25
n(A) = number of trees planted by Parth
n(B) = number of trees planted by Pradnya
n (A ∩ B) = number of trees planted by both of them together
n (A ∪ B) = number of trees planted by Parth or Pradnya
As we know, n (A ∪ B) = n(A) + n(B) - n (A ∩ B)
⇒ n (A ∪ B) = 70 + 90 - 25
⇒ n (A ∪ B) = 160 - 25
⇒ n (A ∪ B) = 35
Question 13.
If n(A) = 20, n(B) = 28 and n (A ∪ B) = 36 then n (A ∩ B) = ?
Answer:
Given: n(A) = 20, n(B) = 28 and n (A ∪ B) = 36
As we know, n (A ∪ B) = n(A) + n(B) - n (A ∩ B)
⇒ n (A ∩ B) = 20 + 28 - 36
⇒ n (A ∩ B) = 48 - 36
⇒ n (A ∩ B)) = 12
Question 14.
In a class, 8 students out of 28 have a dog as their pet animal at home, 6 students have a cat as their pet animal. 10 students have dog and cat both, then how many students do not have a dog or cat as their pet animal at home?
Answer:
Given: n(A) = 8, n(B) = 6 and n (A ∩ B) = 10
n(A) = number of students having only dog as pet
n(B) = number of students having only cat as pet
n (A ∩ B) = number of trees planted by both of them together
Total students having a dog or cat as their pet animal at home = 8 + 6 + 10
= 24
Hence, number of students do not have a dog or cat as their pet animal at
Home = 28 – 24 = 4
Question 15.
Represent the union of two sets by Venn diagram for each of the following.
A = {3, 4, 5, 7}
B = {1, 4, 8}
Answer:
Given: 
Question 16.
Represent the union of two sets by Venn diagram for each of the following.
P = {a, b, c, e, f}
Q = {I, m, n, e, b}
Answer:
Given: 
Question 17.
Represent the union of two sets by Venn diagram for each of the following.
is a prime number between 80 and 100}
is an odd number between 90 and 100}
Answer:
Given: 
is a prime number between 80 and 100}
is an odd number between 90 and 100}
X = {83, 89, 93, 97}
Y = {91, 93, 95, 97, 99}
Question 18.
Write the subset relations between the following sets.
X = set of all quadrilaterals.
Y = set of all rhombuses.
S = set of all squares.
T = set of all parallelograms.
V = set of all rectangles.
Answer:
Given: X = set of all quadrilaterals.
Y = set of all rhombuses.
S = set of all squares.
T = set of all parallelograms.
V = set of all rectangles.
S ⊆ X, because all squares are quadrilaterals.
V ⊆ X, because all rectangles are quadrilaterals.
T ⊆ X, because all parallelograms are quadrilaterals.
S ⊆ Y, because all squares are rhombus.
S ⊆ V, because all squares are rectangles.
S ⊆ T, because all squares are parallelograms.
V ⊆ T, because all rectangles are parallelograms.
Y ⊆ T, because all rhombus are parallelograms.
Question 19.
If M is any set, then write 
and 
Answer:
Given: M is any set
As we know ϕ is null set.
Hence,
(M ∪ ϕ ) = M
(M ∩ ϕ ) = ϕ
Question 20.
Observe the Venn diagram and write the given sets U,A,B,A ∪ B and A ∩ B.
Answer:
(i) U = {1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13}
(ii) A = {1, 2, 3, 5, 7}
(iii) B = {1, 5, 8, 9, 10}
(iv) (A ∪ B ) = {1, 2, 3, 5, 7, 8, 9, 10}
(v) (A ∩ B ) = {1, 5}
Question 21.
If n(A) = 7, n(B) = 13, n (A ∩ B) =4 then n (A ∪ B) = ?
Answer:
Given: 
As we know, n (A ∪ B) = n(A) + n(B) - n (A ∩ B)
⇒ n (A ∪ B) = 7 + 13 - 4
⇒ n (A ∪ B) = 20 - 4
⇒ n (A ∪ B) = 16
