Q1.
If A and B are two sets such that A ∪ B has 20 elements, A has 15 elements and B has 14 elements, how many elements does A
∩ B have ?
(a) 9
(b) 5
(c) 3
(d) 8
Solution. a
As per addition theorem
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
20 = 15 + 14 - n(A ∩ B)
20 = 29 - n(A ∩ B)
n(A ∩ B) = 9
Solution
Q2.
If A and B are two sets such that A has 30 elements , A ∪ B has 45 elements and A ∩ B has 10 elements. Find n(B)
(a) 30
(b) 35
(c) 32
(d) 25
Solution. d
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
45 = 30 + n(B) - 10
55 = 30 + n(B)
n(B) = 25
Solution
Q3.
In an enginearing collage there are 28 teachers who teach mathematics or physics , in which 16 teachers teach mathematics and 3
teachers teach both maths and physics . Find the number of teachers who teach physics ?
(a) 10
(b) 15
(c) 12
(d) 11
Solution. b
Let M denotes maths teachers and P denotes physics teachers.
then as per question
n(M ∪ P) = n(M) + n(P) - n(M ∩ P)
28 = 16 + n(P) - 3
28 = 13 + n(P)
n(P) = 15
So, Physics teachers are 15
Solution
Q4.
If A and B are two sets such that n(A) = 21, n(B) = 27, n(A ∪ B) = 50. Find n(A - B) .
(a) 23
(b) 20
(c) 25
(d) 30
Solution. a
We know that
n(A - B) = n(A ∩ B') = n(A) - n(A ∩ B) = n(A ∪ B) - n(B)
n(A - B) = 50 - 27 = 23
Solution
Q5.
If n(A - B) = 12, n(B - A) = 9, n(A ∪ B) = 25, Find n(A ∩ B) .
(a) 4
(b) 6
(c) 8
(d) 2
Solution. a
As we know that a very important relationship
n(A ∪ B) = n(A - B) + n(B - A) + n(A ∩ B)
25 = 12 + 9 + n(A ∩ B)
25 - 12 = n(A ∩ B)
n(A ∩ B) = 4
Solution
Q6.
If n(U) = 57 , n(A ∪ B) = 42, then find n(A' ∩ B')
(a) 10
(b) 15
(c) 12
(d) 20
Solution.b
We know that
n(A' ∩ B') = n(A ∪ B)' = n(U) - n(A ∪ B)
n(A' ∩ B') = 57 - 42 = 15
Solution
Q7.
In a town 400 car owners investigated, 250 owned car A and 150 owned car B, 60 owned both A and B cars, Is this data correct ?
(a) Yes, data is correct
(b) Data is incorrect
(c) can't say
(d) none of these
Solution. a
Let U be the set of car owners investigated,
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
n(A ∪ B) = 250 + 150 - 60
= 400 - 60 = 340
yes this data is correct
As we know that (A ∪ B) CU
Note : (A ∪ B) is a subset of universal set (U)
Solution
Q8.
If n(U) = 60, n(A) = 45 and n(B) = 20 then find the greatest value of n(A ∪ B) .
(a) 60
(b) 100
(c) 70
(d) 80
Solution. a
It is clear that every set under consideration is a subset of U (the universal set) ∴
A ∪ B C U
n(A ∪ B) ≤ n(U)
n(A ∪ B) ≤ 60
Thus the greatest value of n(A ∪ B) = 60
Solution
Q9.
Let A and B be two sets such that n(A) = 20, n(A ∪ B) = 42 and n(A ∩ B) = 4 .
Find n(A - B) ∪ n(B - A) ?
(a) 35
(b) 38
(c) 30
(d) 32
Solution. b
As we know a very important relationship
n(A - B) ∪ n(B - A) = n(A ∪ B) - n(A ∩ B)
= 42 - 4 = 38
Solution
Q10.
In a survey of 800 students in a town , 190 were listed as drinking tea, 300 drinking coffee and 120 were marked as both drinking tea
and coffee . Find the number of students who drink neither tea nor coffee.
(a) 430
(b) 330
(c) 450
(d) 400
Solution. a
As per question n(∪) = 800
A denotes the set of students like tea
So, n(A) = 190
B denotes the set of students like tea
So, n(B) = 300
n(A ∩ B) = 120 (like both tea & coffee)
Now, we have to find the number of students who drink neither tea nor coffee = n(A'∩B')
As per De-morgan Law ,
n(A'∩B') = n(A ∪B)'
n(A ∪B)' = n(∪) - {n(A ∪ B)}
n(A ∪ B)' = n(∪) - {n(A) + n(B) - n(A ∩ B)}
n(A ∪ B)' = 800 - {190 + 300 - 120}
= 800 - (490 - 120)
= 430
Solution
