Set theory - Maxima & Minima Concepts - Gaurav Sharma

[gaurav_sharma]

From the given Venn diagram, we have

A = a + f + g + d
B = b + f + g + e
C = c + d + g + e

Only A = a, Only B = b, Only C = c
Exactly I = a + b + c
Exactly II = e + d + f
Exactly III = g

We have, A U B U C = I + II + III
And A + B + C = (a + f + g + d) + (b + f + g + e) + (c + d + g + e)
= a + b + c + 2 (d + e +f) + 3(g) = I + 2II + 3III

Guru Mantra
Whole topic revolves around these two equations
I + II + III = A U B U C
I + 2II + 3III = A + B + C

Basic Formulae
(A U B ) = A + B - (A ∩ B )
A U B U C = A + B + C - (A ∩ B ) - ( B ∩ C) - (C ∩ A) + (A ∩ B ∩ C)

In a survey conducted among 400 CAT aspirants, the following points were made
a) 150 students like to study DI-LR
b) 200 students like to study VA
c) 200 students like to study QA
d) 50 students like to study all the three subjects

How many students like studying only 1 subject?
If 120 students like studying VA only, then how many students like to study QA and DI-LR only?
If 40 students like to study QA and DI-LR only, then how many students like to study only VA?
What is the minimum possible number of students who study QA only?

I: only one subject
II: 2 subjects
III: all the three subjects = 50 (Given)
Now, I + II + III = 400
And I + 2II + 3 III = 200 + 200 + 150 = 550
Also III = 50
So on solving the two equations, we get II = 50 and I = 300
So 300 students like studying only 1 subject.

We can represent the given data in Venn diagram as shown

Now, 200 = a + c + 50 + 120 = > a + c = 30
Also, II = a + c + b = 50
b + 30 = 50
b = 20
Hence students who like study QA and DI-LR only are 20

We can represent the given data in a Venn diagram as shown

40 students like to study QA and DI-LR only
II = a + c + 40
a + c = 10
only VA = 200 – (50 + a + c)
= 200 – 60
= 140

We have to find the minimum possible number of students who study QA only
We have to maximize a and b
Also, a + b + c = 50
Maximum value of a + b can be 50 if c = 0

Hence, students who like to study only QA = 200 – (50 + 50) = 100

A survey of 200 people in a community who watched at least one of three channels B, C and D showed that 80% of the people watched D, 22% watched B and 15% watched C. If 5% watched D and C, 10% watched D and B then what percentage of people watched B and C only?
a) 2
b) 4
c) Cannot be determined
d) 8

Solving the questions by percentages

We have I + II + III = 100
And I + 2II + 3 III = 80 + 22 + 15 = 117
II + 2 III = 17
a + g + h + 2e = 17
And we have e + g = 5, a + e = 10
a + g + 2e = 15
h = (a + g + h + 2e) – (a + g + 2e)
h = 17 – 15 = 2%
h = 2% of 200 = 4

In a business school there are 3 electives offered to the students, where students have a choice of not choosing any electives. 70% of students opted for marketing, 60% students opted for HR and 50% students opted for Finance. 30% students opted for Marketing and HR both, 40% opted for HR and finance both, 20% opted for Finance and Marketing both.
a) What is the minimum number of students opted for all 3 electives?

We have I + II + III less than or equal to 100
I + 2II + 3III = 70 + 50 + 60 = 180
II + 3III = 30 + 40 + 20 = 90
We have to minimize III
Solving the equations, we get
I + II = 90
Putting III = 0 in II + III = 90, we get II = 90
And putting II = 90 in I + II = 90, we get I = 0
So I = 0, III = 0 and II = 90 satisfies all the conditions.
Hence minimum number of students who opted for all 3 subjects are 0

85 children went to an amusement park where they could ride on the merry-go-round, roller coaster and giant wheel. It was known that 20 of them took all three rides and 55 of them took at least 2 of the 3 rides. Each ride costs rupee 1 and total expense is 145.
a) How many children did not try any of the rides?
b) How many took exactly one ride?

We have III = 20
And II + III = 55
II = 55 – 20 = 35
Total expense is 145
I + 2II + 3III = 145
I + 2 x 35 + 3 x 20 = 145
I = 15
So, I + II + III = 15 + 35 + 20 = 70
Number of students who did not take any of the rides = 85 – 70 = 15
Number of students who took exactly 1 ride = I = 15

In a business school there are 3 electives and at least one elective is compulsory to opt. 75% students opted for Marketing, 62% students opted for Finance and 88% students opted for HR. A student can have dual or triple specialization. What is the maximum number of students that can specialize in all three stream?

We have I + II + III = 100 and I + 2II + 3III = 225
We have to maximize III, so we have to minimize II
Put II = 0
I + III = 100 and I + 3III = 225
This does not gives us integral value for I and III
Put II = 1
We get I + III = 99 and I + 3III = 223
I = 37, II = 1 and III = 62
Hence Maximum value of III = 62

Practice Questions

Q1) In a group of 1000 people, 700 can speak English and 500 can speak Hindi. If all the people speak at-least one of the two languages, find
a) How many can speak both the languages?
b) How many can speak exactly one language?

Q2) In an exam, 60% of the candidates passed in maths and 70% candidates passed in English and 10% candidates failed in both the subjects. If 300 candidates passed in both the subjects find the total number of candidates appeared in the exams, if they took test in only two subjects viz Maths & English.

Q3) In a car agency one day 120 cars were decorated with three different accessories via, power window, AC and music systems. 80 cars were decorated with power windows, 84 cars were decorated with AC and 80 cars were decorated with music systems. What is the minimum and maximum number of cars which were decorated with all of three accessories?
a) 10, 61
b) 10, 45
c) 25, 35
d) None of these

Q4) A group of 100 people play carom, snooker and chess. 90 play chess, 80 play carom and 80 play snooker. Find the maximum number of people who play all 3 games, if each person plays at least one game
a) 80
b) 70
c) 75
d) 65

Q5) In a society with 200 people, 77 people are dog owners, 80 are cat owners and 73 are bird owners. If 40 people do not have any of these pets, what is the maximum possible number of people who own all three pets?

Q6) A survey about preferred TV channels was conducted among a group of 10000 people. The following were the results – 93% liked Sony TV, 89% liked Zee TV, 81% liked Star plus, 75% liked Zee Cinema, 78% liked MTV and 100 people did not like any of these 5 channels. Find the minimum number of people who like all these five channels
a) 1500
b) 1000
c) 2500
d) 2000