  # Set theory questions for CAT: MBA Preparation

Set theory questions is one of the interesting topics CAT aspirants enjoy solving them. However, lack of conceptual knowledge in this topic might lead to confusion. Before attempting any question from this topic one must be clear with all Set theory laws and Venn diagrams. Set complement, intersection, Union and universal set questions are of prime importance.

# SET THEORY QUESTIONS

#### Question 1)

Set Fn gives all factors of n. Set Mn gives all multiples of n less than 1000. Which of the following statements is/are true?
a. F108 ∩ F84 = F12
b. M12 ∪ M18 = M36
c. M12 ∩ M18 = M36
d. M12 ⊂ (M∩ M4)

1. a, b, and c only
2.  a, c and d only
3. a and c only
4.  All statements are true.

Answer: 2) a, c, and d only.

#### Explanation 1.

i. F108 ∩ F84
This is the set of all numbers that are factors of both 108 and 84
=> this is set of all common factors of 84 and 108
=> this is set of numbers that are factors of the Highest Common Factor of 84 and 108.

HCF (84, 108) = 12.

F108 ∩ F84 = F12 – this is true.

b)M12 will have numbers {12, 24, 36, 48, ….} Numbers like 12, 24, … will not feature in M36. So, Statement B cannot be true.

c) M12 ∩ M18 – this is the set of all common multiples of 12 and 18.
This is the set of numbers that are multiples of the Least Common Multiple of 12 and 18. (which is 36).

This statement is also true. iv. Using the same logic as that used in statement iii, we can determine that statement iv is also true. Remember that every set is a subset of itself.

#### Question 2)

Set P comprises all multiples of 4 less than 500. Set Q comprises all odd multiples of 7 less than 500, Set R comprises all multiples of 6 less than 500. How many elements are present in P∪Q∪R?

1. 202
2. 243
3. 228
4. 186

#### Explanation 2.

This question involves both Number Systems and Set Theory.

Set P = {4, 8, 12, ….496} ↦ 124 elements {all elements from 1 x 4 to 124 x 4}
Set Q = {7, 21, 35, 49,……497} ↦ {7 × 1, 7 × 3, 7 × 5, ….. 7 × 71} ↦ 36 elements.
Set R = {6, 12, 18, 24, …..498} ↦ {6 × 1, 6 × 2, 6 × 3, ….. 6 ×83} ↦ 83 elements.

Sets P and R have only even numbers; set Q has only odd numbers. So,
P∩Q = Null set
Q∩R = Null set
P∩Q∩R = Null set
So, If we find P∩R, we can plug into the formula and get P∪Q∪R

P∩R = Set of all multiples of 12 less than 500 = {12, 24, 36,…..492 = {12 × 1, 12 × 2 , 12 × 3, …12 ×41} ↦ This has 41 elements
P∪Q∪R = P + Q + R – (P∩Q) – (Q∩R) – (R∩P) + (P∩Q∩R)
P∪Q∪R = 124 + 36 + 83 – 0 – 0 – 41 + 0 = 202

#### Question 3)

In a survey it was found that 10% people don’t use Facebook, Twitter or Whatsapp.8% uses all the three. There are 15% who uses Facebook and Twitter, 20% who use Twitter and Whatsapp and 20% who useFacebook and Whatsapp. The number of people that use only Facebook, only Twitter and only Whatsapp is equal. If the survey was conducted on 1000 people, answer the following :
What is the ratio of a number of people that uses Whatsapp only to the people using either Whats app or Facebook or both?

1. 1/6
2. 25/75
3. 1/9
4. 1/3

#### Explaination 3.

Let the no. of people that use only Facebook = only Twitter = only Whatsapp = x
Let’s construct of Venn diagram: From the figure we can see that x + x + x + 8 + 20 + 20 + 15 + 10 = 100
=) 3x = 100 – 73
=) x = 27/3 = 9

Thus the of no people who use Whatsapp only = 9% or 9 x 1000/100 = 90

% of people that use Whatsapp only = 9
% of people that use either Whatsapp or Facebook or both = 20 + 20 + 8 + 9 + 9 +15 = 81 Thus the ratio = 9/81=1/9

#### Question 4)

In a class 40% of the students enrolled for Math and 70% enrolled in Economics. If 15% of the students enrolled for both Math and Economics, what % of the students of the class did not enroll for either of the two subjects?

1. 5%
2. 15%
3. 0%
4. 25%

#### Explanation 4.

We know that (A U B) = A + B – (A n B), where (A U B) represents the set of people who have enrolled for at least one of the two subjects Math or Economics and (A n B) represents the set of people who have enrolled for both the subjects Math and Economics.

Note
(A U B) = A + B – (A n B) => (A U B) = 40 + 70 – 15 = 95%

That is 95% of the students have enrolled for at least one of the two subjects Math or Economics.
Therefore, the balance (100 – 95)% = 5% of the students have not enrolled for either of the two subjects.

#### Question 5)

A shop sells three type of products i.e. Pen, Pencil & Notebook. On a survey for checking the sales of each product of the shop he found that the no. of people who bought only pen, only pencil, & only notebook are in A.P. in no particular order. Similarly, the number of people who bought exactly two of the three products are in A.P. too.
It was also found that the no. of people who bought all the products is 1/20th of the number of people who bought pencil only which in turn is equal to half of the number of people who bought notebook only. The number of people that bought both pen & pencil is 15, whereas that of those who bought pencil & notebook is 19. The number of people who bought notebook are 120, which is more than the no. of people who bought pen (which is a 2 digit no above 50).

What is the total no people that visit the shop?

1. 220
2. 231
3. 233
4. 240

#### Explanation 5.

Let the no of people that bought all the three products be ‘x’, and no of people that bought both pen and notebook be y .
Then no of people that bought pencil only = 20 x
And those who bought notebook only = 40 x
Venn diagram according to question can be drawn as: Thus from the figure we get 40x + (19 -x) + x + y = 120
=) 40x + y = 101
Now if we take a close look at the equation we can conclude that the value x can be either ‘1’ or ‘2’
Putting x = 1 in the equation we get y =61. But we have to also meet the additional condition that 15- x, 19-x, y are in A.P. which is not possible in this case. Hence it is rejected.
Thus the value of x = 2, hence we will get y = 21, which also satisfies the additional condition.
Now if we see the figure again we can see that only pen can either be 0, 60, and 120 to satisfy the additional condition. And as it cannot be less than 50 (Specified in the question), the only value only pencil can take = 60
Thus, Total no of people that visited the shop = 60 + 13 + 2 + 40 + 17 + 80 + 21 = 233

There are n numbers of Set theory questions available online. But one need to do smart work with hard work ie to practice many questions but of different types. Finally, you will be prepared for this topic.