Set Theory Questions for CAT with Answers PDF

A = {1, 4, 9, 16}
B = {2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20}
C = A ∪ B and D = A ∩ B = {1, 2, 3, … 19, 20}
D = Ø
(i) (C ∪ D) ∪ D = A ∪ B
C ∪ D = C
C = C
(ii) C ∩ D = Ø
Ø = Ø
(ii) C ∪ D = C
C = C
All the statements are true.

Let us consider that the number of people in the locality be 30.
Number of persons having Cable TV 
Number of persons having VCR 
Number of persons having both 
Number of persons having either Cable TV or VCR = 20 + 6 – 3 = 23 
Required fraction  = 23/30

A total of 15 persons subscribed to at least two of these magazines and 2 persons subscribed to all the three.
Then, 13 subscribed to exact two of these magazines.
Now, n(A) + n(B) + n(C) - [n(A ∩ B) + n(B ∩ C) + n(C ∩ A)] + n(A ∩ B ∩ C) = 39 + 26 + 6 - (13 + 2 + 2 + 2) + 2 = 54
Thus, out of 100 people surveyed, 54 subscribed to at least one of the three magazines or 46 did not subscribe to any of the three.

From the given data, number of people eating vegetarian food as well as mutton cannot be determined.

Number of people who eat only vegetarian food = 250
Number of people who eat non-vegetarian food = 600
Number of people who eat mutton or chicken = 100 + 200 + 200 = 500
Remaining number of non-vegetarian people who do not eat mutton or chicken = 600 - 500 = 100
So, number of people who do not eat chicken or mutton = 250 + 100 = 350
 

It is given that 24 were either men or dentists. We know that 23 were men, so the remaining 1 must be a woman dentist.

Putting the values of n = 1, 2, 3, 4, …, we get:
X = 0, 9, 54, …
Y = 0, 9, 18, 27, 36, 45, 54, ...
Hence, X U Y will be a set equal to the value of Y.
Hence, X U Y = Y

C = Cricket
T = Tennis

Total number of people = 65
Let x be the number of people who like only tennis.
Number of people who like cricket = 40
Number of people both cricket and tennis = 10
From the Venn diagram,
40 - 10 + 10 + x = 65
x = 25
i.e. Number of people who like only tennis = 25
Number of people who like tennis = 25 + 10 = 35

(A × B) ∩ (B × A) will be an empty set if A and B are disjoint. This is because in (A × B) ∩ (B × A), when we use any element, it means we are finding the common elements. But A and B are disjoint sets, so we will get zero common elements. Hence, the intersection of (A × B) and (B × A) will be a null set.

This problem can be solved with the help of a Venn diagram.

Number of men belonging to the middle class = 240 × (1/2) = 120
Number of men belonging to the upper class = 240 × (1/3) = 80
Number of men belonging to both the classes = 240 × (1/4) = 60
We have identified 60 + 60 + 20 = 140 men as having well-defined class.
∴ 240 – 140 = 100 men do not belong to either class.