🎉Ace the Final Stretch with our Last Mile Excellence – Your Ultimate CAT 2024 Boost!

CAT Questions | CAT Set theory

CAT Quantitative Aptitude | CAT Set theory, Union and Intersection,

A CAT Set Theory Problems ; Set theory and Venn Diagrams are one of the most commonly tested topics in CAT exam. Questions from CAT Set theory have appeared consistently in the CAT exam for the last several years. Set theory and Venn diagram is a very interesting topic, as it is also useful in CAT Data Interpretation and Logical Reasoning . CAT exam does not only check for formulaic knowledge in this idea, but also for strong fundamentals and application of the concepts involved. One can usually expect 1~2 questions from Set Theory in the CAT exam. Make use of 2IIMs Free CAT Questions, provided with detailed solutions and Video explanations to obtain a wonderful CAT score. If you would like to take these questions as a Quiz, head on here to take these questions in a test format, absolutely free.

CAT Set Theory - Number Theory

Set F_n gives all factors of n. Set M_n gives all multiples of n less than 1000. Which of the following statements is/are true?

i. F₁₀₈ ∩ F₈₄ = F₁₂
ii. M₁₂ ∪ M₁₈ = M₃₆
iii.M₁₂ ∩ M₁₈ = M₃₆
iv.M₁₂ ⊂ M₆ ∩ M₄
1. i, ii and iii only
2. i, iii and iv only
3. i and iii only
4. All statements are true
Choice B
Statements i, iii and iv only
CAT Set Theory - De Morgan's Laws

A´ is defined as the complement of A, as in, set of all elements that are part of the universal set but not in A. How many of the following have to be true?

i. {n(A ∪ B)' =n(A' ∩ B')}
ii. If {A ∩ B=0}, then {A' ∪ B'} is equal to the universal set
iii. If {A ∪ B} = universal set, then {A' ∩ B'} should be the null set.
iv. If {A ⊂ B} then {A' ∪ B'=(A ∩ B)'}
1. 1
2. 2
3. 3
4. 4
Choice D
all four statements are true.
CAT Set Theory: Venn Diagrams

Of 60 students in a class, anyone who has chosen to study maths elects to do physics as well. But no one does maths and chemistry, 16 do physics and chemistry. All the students do at least one of the three subjects and the number of people who do exactly one of the three is more than the number who do more than one of the three. What are the maximum and minimum number of people who could have done Chemistry only?
1. 40, 0
2. 28, 0
3. 38, 2
4. 44, 0
Choice D
44 and 0 respectively
CAT Calendars - Counting Birthdays

John was born on Feb 29^th of 2012 which happened to be a Wednesday. If he lives to be 101 years old, how many birthdays would he celebrate on a Wednesday?
1. 3
2. 4
3. 5
4. 1
Choice B
4
CAT Calendars - Recurring Days

How many of the following statements have to be true?

i. No year can have 5 Sundays in the month of May and 5 Thursdays in the month of June.
ii. If Feb 14^th of a certain year is a Friday, May 14^th of the same year cannot be a Thursday
iii. If a year has 53 Sundays, it can have 5 Mondays in the month of May.
1. 0
2. 1
3. 2
4. 3
Choice B
1
CAT Set Theory: Number Of Elements

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
Choice A
202
CAT Set Theory: Min and Max % of people

95% of the students in a class have taken Marketing, 80% have chosen Finance, 84% have chosen operations (ops), and 90% have chosen Human Resources (HR). What is the maximum and minimum percentage of people who have chosen all of the four?
1. 80% and 56%
2. 95% and 53%
3. 80% and 49%
4. 80% and 51%
Choice C
80% and 49%
CAT Set Theory: Number of Elements

Set A comprises all three digit numbers that are multiples of 5, Set B comprises all three–digit even numbers that are multiples of 3 and Set C comprises all three–digit numbers that are multiples of 4. How many elements are present in {A ∪ B ∪ C}?
1. 420
2. 405
3. 555
4. 480
Choice A
420
CAT Set Theory and Combinatorics

Set A = {2, 3, 5, 6, 7}, Set B = {a, b, c}. How many onto functions can be defined from Set B to Set A?
1. 2
2. 3
3. 4
4. None of the above
Choice D
None of the above
CAT Set Theory: Sets and Unions

Sonu started a new business with accounts in two different banks (i.e. Axis and SBI).He deposited the earnings of each day in either of the two banks. However he does not deposit his earnings in both the banks simultaneously on any given day. However somehow he could not carry the business for long and had to shut it down. Find the total no of days Sonu carried on the business if…

1) He did not deposited in axis bank on 20 days and in SBI on 24 days.
2) He deposited on either axis bank or SBI on 28 days.
1. 36
2. 18
3. 13
4. 24
Choice A
36
CAT Set Theory: Sets and Unions

A class in college has 150 students numbered from 1 to 150 , in which all the even numbered students are doing CA, whose number are divisible by 5 are doing Actuarial and those whose numbers are divisible by 7 are preparing for MBA. How many of the students are doing nothing?
1. 37
2. 45
3. 51
4. 62
Choice C
51
CAT Set Theory: Sets and Unions

In a class of 345 students, the students who took English, Math and Science are equal in number. There are 30 students who took both English and Math, 26 who took both Math and Science, 28 who took Science and English and 14 who took all the 3 subjects.There are 43 students who didn’t take any of the subjects. Answer the following question according to the data given above.
How many students have taken English as a subject?
1. 286
2. 124
3. 246
4. 108
Choice B
124
CAT Set Theory: Sets and Unions

In a class of 345 students, the students who took English, Math and Science are equal in number. There are 30 students who took both English and Math, 26 who took both Math and Science, 28 who took Science and English and 14 who took all the 3 subjects.There are 43 students who didn’t take any of the subjects. Answer the following question according to the data given above.
How many students have taken only one subject?
1. 286
2. 124
3. 246
4. 108
Choice C
246
CAT Set Theory: Sets and Unions

In a class of 345 students, the students who took English, Math and Science are equal in number. There are 30 students who took both English and Math only, 26 who took both Math and Science only, 28 who took Science and English only and 14 who took all the 3 subjects.There are 43 students who didn’t take any of the subjects. Answer the following question according to the data given above.
What percent of students did not take Science?
1. Less than 55%
2. approx. 59%
3. 72%
4. 79%
Choice B
approx. 59%
CAT Set Theory: Sets and Unions

In a survey conducted to know people’s preference for android phones and I phones, 80 person preferred android phones while 60 person preferred I phones. There were 20 who liked both and may prefer any. If there was no one who didn’t prefer at least one of the phones,then on how many people was the survey conducted?
1. 120
2. 40
3. 80
4. 60
Choice A
120
CAT Set Theory: Sets and Unions

In Grand Oberoi hotel, 1160 guests are present currently. The hotel provides the following extra facilities: Gym, Swimming, Fun park, Food. During a regular survey the management team of Oberoi noticed something quite extraordinary about the extra facilities provided by them. They noticed that for every person who uses ‘F’ no. of facilities, there are exactly 3 persons who uses at least (F-1) no. of facilities, F = 2, 3, 4. They also found that the no. of persons who used no extra facilities is twice the no of person that used all the 4 facilities. Help the management team to find out how many persons used exactly 3 facilities.
1. 40
2. 60
3. 80
4. 100
Choice C
80
CAT Set Theory: Sets and Unions

In its annual fest, a college is organizing three events: B-quiz, Finance and Marketing. The college has a strength of 510 students.The students were allowed to participate in any no. of events they liked. While viewing the statistics of the performance, the general secretary noticed:-
1. The number of students who participated in atleast two events were 52% more than those who participated in exactly one game.
2. The no. of students participating in 1,2 or 3 events respectively was atleast equal to 1.
3. The number of students who did not participate in any of the three events was the minimum possible integral value under these conditions.
What can be the maximum no. of students who participated in exactly 3 games?
1. 200
2. 300
3. 303
4. 304
Choice C
303
CAT Set Theory: Sets and Unions

A factory has 80 workers and 3 machines. Each worker knows to operate atleast 1 machine. If there are 65 persons who knows to operate machine 1, 60 who knows to operate machine 2 and 55 who knows to operate machine 3,what can be the minimum number of persons who knows to operate all the three machines?
1. 15
2. 20
3. 30
4. 40
Choice B
20
CAT Set Theory: Sets and Unions

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 use Facebook and Twitter only, 20% who use Twitter and Whatsapp only and 20% who use Facebook and Whatsapp only. Number of people that use only Facebook, only Twitter and only Whatsapp are equal. If the survey was conducted on 1000 people, answer the following:
How many people use Whatsapp only?
1. 20
2. 9
3. 15
4. 90
What is the ratio of number of people that uses Whatsapp only to the people using either Whats app or Facebook or both?
1. \\frac{1}{6}\\)
2. \\frac{25}{75}\\)
3. \\frac{1}{3}\\)
4. \\frac{1}{9}\\)
90 and \\frac{1}{9}\\)
CAT Set Theory: Sets and Unions

In class of 280 students, each student needs to choose between the three extra subject (i.e IT, Hindi and Sanskrit) offered along with the course. The students that choose each of these subjects are 160, 130, 110. The number of students who choose more than one of the three is 40% more than the number of students who choose all the three subjects If there are no students who choose none of the 3 subjects, how many students study all the three subjects?
1. 40
2. 50
3. 80
4. 100
Choice B
50
CAT Set Theory: Sets and Unions

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 \\frac{1}{20}\\)^th 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 visits the shop?
1. 220
2. 231
3. 233
4. 240
How many people bought both pen and notebook?
1. 15
2. 17
3. 19
4. 21
233, 21
CAT Set Theory: Sets and Unions

In a survey it was found that, the number of people that like only Pepsi, only Coke, both Coke and Pepsi and neither of them are 2n, 3n, \\frac{69}{n}\\), \\frac{69}{3n}\\) respectively. What is the no of people who drink coke?
1. 72
2. 49
3. Both a and b
4. cannot be determined
Choice A
72

The Questions that follow, are from actual CAT papers. If you wish to take them separately or plan to solve actual CAT papers at a later point in time, It would be a good idea to stop here.

CAT 2022 Slot 1 - QA

In a class of 100 students, 73 like coffee, 80 like tea and 52 like lemonade. It may be possible that some students do not like any of these three drinks. Then the difference between the maximum and minimum possible number of students who like all the three drinks is
1. 48
2. 53
3. 47
4. 52
Choice C
47
CAT 2020 Question Paper Slot 2 - Set Theory

Students in a college have to choose at least two subjects from chemistry, mathematics and physics. The number of students choosing all three subjects is 18, choosing mathematics as one of their subjects is 23 and choosing physics as one of their subjects is 25. The smallest possible number of students who could choose chemistry as one of their subjects is
1. 22
2. 19
3. 20
4. 21
CAT 2019 Question Paper Slot 1 - Set Theory

A club has 256 members of whom 144 can play football, 123 can play tennis, and 132 can play cricket. Moreover, 58 members can play both football and tennis, 25 can play both cricket and tennis, while 63 can play both football and cricket. If every member can play at least one game, then the number of members who can play only tennis is
1. 32
2. 43
3. 38
4. 45
Choice B
43
CAT 2018 Question Paper Slot 2 - Set Theory

For two sets A and B, let AΔB denote the set of elements which belong to A or B but not both. If P = {1,2,3,4}, Q = {2,3,5,6,}, R = {1,3,7,8,9}, S = {2,4,9,10}, then the number of elements in (PΔQ)Δ(RΔS) is
1. 7
2. 8
3. 9
4. 6
Choice A
7
CAT 2018 Question Paper Slot 2 - Set Theory

If A = {6²ⁿ - 35n - 1: n = 1,2,3,...} and B = {35(n-1) : n = 1,2,3,...} then which of the following is true?
1. Neither every member of A is in B nor every member of B is in A
2. Every member of A is in B and at least one member of B is not in A
3. Every member of B is in A.
4. At least one member of A is not in B
Choice B
Every member of A is in B and at least one member of B is not in A
CAT 2018 Question Paper Slot 1 - Set Theory

Each of 74 students in a class studies at least one of the three subjects H, E and P. Ten students study all three subjects, while twenty study H and E, but not P. Every student who studies P also studies H or E or both. If the number of students studying H equals that studying E, then the number of students studying H is [TITA]
52
CAT 2018 Question Paper Slot 1 - Set Theory

If among 200 students, 105 like pizza and 134 like burger, then the number of students who like only burger can possibly be
1. 26
2. 23
3. 96
4. 93
Choice D
93

The Questions that follow, are from actual XAT papers. If you wish to take them separately or plan to solve actual XAT papers at a later point in time, It would be a good idea to stop here.

XAT 2020 Question Paper - QADI
Nalini has received a total of 600 WhatsApp messages from four friends Anita, Bina, Chaitra and Divya. Bina and Divya have respectively sent 30% and 20% of these messages, while Anita has sent an equal number of messages as Chaitra. Moreover, Nalini finds that of Anita’s, Bina’s, Chaitra’s and Divya’s messages, 60%, 40%, 80% and 50% respectively are jokes. What percentage of the jokes, received by Nalini, have been sent neither by Divya nor by Bina?
1. 65.12
2. 61.4
3. 57
4. 38.6
5. 34.88
Choice B
61.4
XAT 2020 Question Paper - QADI
Read the information given below and answer the 2 associated questions.
190 students have to choose at least one elective and at most two electives from a list of three electives: E1, E2 and E3. It is found that the number of students choosing E1 is half the number of students choosing E2, and one third the number of students choosing E3.
Moreover, the number of students choosing two electives is 50.
Which of the following CANNOT be obtained from the given information?
1. Number of students choosing exactly one elective
2. Number of students choosing E1
3. Number of students choosing E3
4. Number of students choosing both E1 and E2
5. Number of students choosing either E1 or E2 or both, but not E3
Choice DNumber of students choosing both E1 and E2
XAT 2020 Question Paper - QADI
In addition to the given information, which of the following information is NECESSARY and SUFFICIENT to compute the number of students choosing only E1, only E2 and only E3?
1. Number of students choosing both E1 and E2
2. Number of students choosing only E1, and number of students choosing both E2 and E3
3. Number of students choosing only E2, and number of students choosing both E2 and E3
4. No extra information is necessary
5. Number of students choosing both E1 and E2, number of students choosing both E2 and E3, and number of students choosing both E3 and E1
Choice C
Number of students choosing only E2, and number of students choosing both E2 and E3
XAT 2018 Question Paper - QADI
The number of boys in a school was 30 more than the number of girls. Subsequently, a few more girls joined the same school. Consequently, the ratio of boys and girls became 3:5. Find the minimum number of girls, who joined subsequently.
1. 31
2. 51
3. 52
4. 55
5. Solution not possible
Choice C
52
XAT 2018 Question Paper - QADI
In the final semester, an engineering college offers three elective courses and one mandatory course. A student has to register for exactly three courses: two electives and the mandatory course. The registration in three of the four courses is: 45, 55 and 70. What will be the number of students in the elective with the lowest registration?
1. 35
2. 40
3. 42
4. 45
5. Either B or D
Choice E
Either B or D

The Questions that follow, are from actual IPMAT papers. If you wish to take them separately or plan to solve actual IPMAT papers at a later point in time, It would be a good idea to stop here.

IPMAT 2020 Question Paper - IPM Indore Quants
Out of 80 students who appeared for the school exams in Mathematics (M), Physics (P) and Chemistry (C), 50 passed M , 30 passed P and 40 passed C. At most 20 students passed M and P at most 20 students passed P and C and at most 20 students passed C and M. The maximum number of students who could have passed all three exams is __________.
20
IPMAT 2019 Question Paper - IPM Indore Quants
Let the set = {2,3,4,..., 25}. For each k ∈ P, define Q(k)= {x ∈ P such that x > k and k divides x}. Then the number of elements in the set \ P - U_{k=2}^{25} \\) Q(k) is
9
IPMAT 2019 Question Paper - IPM Indore Quants
In a class of 65 students 40 like cricket, 25 like football and 20 like hockey. 10 students like both cricket and football, 8 students like football and hockey and 5 students like all three sports. If all the students like at least one sport, then the number of students who like both cricket and hockey is
1. 7
2. 8
3. 10
4. 12
Choice A
7