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CAT Questions | CAT Algebra Questions: Set Theory Questions

Importance of Set Theory in Various Exams

Set theory or venn diagram is a topic which falls under the logical reasoning side of any entrance exam. In these sets of questions, inter-mingled situations are given out of which repetitive situations need to be identified & eliminated in order to find the missing data. 

Set theory and Venn Diagrams are one of the most commonly tested topics in MBA exams. Questions from the Set theory have appeared consistently in the CAT, XAT, GMAT exam for the last several years. Set theory and Venn diagram is a very interesting topic, as it is also useful in Data Interpretation and Logical Reasoning

Set Theory
YearSlotNo of QuestionsGood attemptDifficulty
2019
Slot 142Moderate
Slot 232Difficult
2018
Slot 144Easy
Slot 243Moderate
2017
Slot 143Easy
Slot 243Easy

XAT Exam:

YearNo of QuestionsGood attemptDifficulty
201922Moderate
201811Easy
201732Moderate-Difficult

IIFT Exam:

YearNo of QuestionsGood attemptDifficulty
201932Moderate
201811Moderate
201721Difficult

Important Concepts in Set Theory

1. What is the union of sets? 

The union of two sets is a set containing all elements that are in AA or in BB (possibly both). For example, {1,2}∪{2,3}={1,2,3}{1,2}∪{2,3}={1,2,3}. Thus, we can write x∈(A∪B)x∈(A∪B) if and only if (x∈A)(x∈A) or (x∈B)(x∈B).

2. What is the intersection of sets?

The intersection of two sets AA and BB, denoted by A∩BA∩B, consists of all elements that are both in AA and−−−and_ BB. For example, {1,2}∩{2,3}={2}{1,2}∩{2,3}={2}.

3. What is the complement of set?

The complement of a set AA, denoted by AcAc or A¯A¯, is the set of all elements that are in the universal set SS but are not in AA.

4. What is disjoint sets?

Two sets AA and BB are mutually exclusive or disjoint if they do not have any shared elements, i.e., their intersection is the empty set, A∩B=∅A∩B=∅. More generally, several sets are called disjoint if they are pairwise disjoint, i.e., no two of them share a common element. 

CAT Quantitative Aptitude questions | CAT Algebra: Set Theory

1. There are 30 students in a class. Among them, 8 students are learning both English and French. A total of 18 students are learning English. If every student is learning at least one language, how many students are learning French in total? (XAT 2018)

The Venn diagram for this problem looks like this.


 

Every student is learning at least one language. Hence there is no one who fall in the category‘neither’.

So in this case, n(EᴜF) = n(µ).

It is mentioned in the problem that a total of 18 are learning English. This DOES NOT mean that 18 are learning ONLY English. Only when the word ‘only’ is mentioned in the problem should we consider it so.

Now, 18 are learning English and 8 are learning both. This means that 18 – 8 = 10 are learning ONLY English.

n(µ) = 30, n(E) = 10

n(EᴜF) = n(E) + n(F) – n(E∩F)

30 = 18+ n(F) – 8

n(F) = 20

Therefore, total number of students learning French = 20.

 

2. 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? (CAT 2019, Slot 1)

a. F108 ∩ F84 = F12

b. M12 ∪ M18 = M36

c. M12 ∩ M18 = M36

d. M12 ⊂ (M6 ∩ M4)

• a, b, and c only

• a, c and d only

• a and c only

• All statements are true

 

 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

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

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.

 

3. 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? (SNAP 2019)

a) 5%

b) 15%

c) 0%

d) 25%

 

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.

 

(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.

 

4. 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 use Facebook 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 Whatsapp or Facebook or both? (CAT 2019, Slot 2)

a) 1/6

b) 25/75

c) 1/9

d) 1/3

 

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

 

How to Deal with Set Theory

Level 1

• List out the formula related to the topic and focus on understanding the fundamentals and application. For this, we recommend using the MBAP CAT E-Book (Concept Theory) study material.

• Break down the steps and simplify the process to understand a question. Also, practice almost 50-60 questions each for a particular concept. If you find difficulty understanding the concepts, use the MBAP Live Lecture Recording (Basic) on Basic Concepts.

• Always look for a definite answer. To get better clarity on Set Theory, use MBAP CAT E-Book (Practice Questions) and solve all the relevant practice questions. Also, solve the MBAP lecture assignment to complete the basics.

Level 2

• Watch tutorials for a better understanding.

• Practicing is the only way to master the Set Theory topic.

• Understand the concepts and learn how to apply them in different applications.

• It will be way easier for you to attempt the questions by solving previous year papers. They are all readily available if you use MBAP Topic Wise Previous Year CAT Questions.

 

Level 3

Students need to look into problems that use multiple concepts to find a solution. Try to answer questions from the advanced level of previous CAT papers, mock tests, and materials. And try to attempt these questions in a time-based manner. Taking too much time for advanced questions will not be considered good to crack the exam. Finally, Practice is the only key to success. For further advancement on set theory, use MBAP CAT Advance E-Books and solve the MBAP Previous Year CAT paper.

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