Complete Exercise 1.1 Solutions - Set Theory and Venn Diagrams
Table of Contents
1. Cardinality of Sets
a) Present the cardinality of sets with examples and show it to your teacher.
Cardinality of a set is the number of elements in the set, denoted by n(A).
Examples:
If A = {1, 2, 3, 4, 5}, then n(A) = 5
If B = {a, b, c, d}, then n(B) = 4
If C = {x | x is a vowel in English alphabet}, then n(C) = 5
b) For two sets A and B, A ⊂ B, find the values of n(A∪B) and n(A∩B).
If A ⊂ B (A is a proper subset of B), then:
All elements of A are in B, so A∪B = B and A∩B = A
Therefore: n(A∪B) = n(B) and n(A∩B) = n(A)
c) If A and B are overlapping sets, state the formula for n(A∪B).
For overlapping sets A and B:
n(A∪B) = n(A) + n(B) - n(A∩B)
d) There are 12 and 8 elements in the sets A and B respectively. Find the minimum number of elements that would be in the set n(A∪B).
Given: n(A) = 12, n(B) = 8
Minimum n(A∪B) occurs when A and B have maximum overlap
Maximum possible n(A∩B) = min(n(A), n(B)) = min(12, 8) = 8
Therefore, minimum n(A∪B) = n(A) + n(B) - n(A∩B) = 12 + 8 - 8 = 12
2. Venn Diagram Analysis
Given: n(M) = 80, n(E) = 90, n(M∪E)' = 15
Let's define the terms:
(a) n₀(M) = 20
(b) n₀(E) = 30
(c) n(M) = n₀(M) + n(M∩E) = 80
(d) n(E) = n₀(E) + n(M∩E) = 90
(e) n(M∪E) =n₀(M) + n₀(E) + n(M∩E) =110
(f) n(M∩E) =60
(g) n(M∪E)' = 15 (elements in neither M nor E)
(h) n(U) = n(M∪E) + n(M∪E)'= 110 + 15 = 125
information about the relationship between M and E is 100% accurate.
Note: The question seems to have sufficient information to determine all values precisely with additional constraints or a diagram showing the specific relationships.
3. Set Calculations
a) If n(U) = 200, n₀(M) = 2x, n₀(E) = 3x, n(M∩E) = 60 and n(M∪E)' = 40, find the value of x.
n(M∪E) = n(U) - n(M∪E)' = 200 - 40 = 160
Also, n(M∪E) = n₀(M) + n₀(E) + n(M∩E)
160 = 2x + 3x + 60
160 = 5x + 60
5x = 100
x = 20
b) If n(U) = 350, n(A) = 200, n(B) = 220 and n(A∩B) = 120, then find n(A∪B) and n(A∪B)'.
n(A∪B) = n(A) + n(B) - n(A∩B) = 200 + 220 - 120 = 300
n(A∪B)' = n(U) - n(A∪B) = 350 - 300 = 50
c) If n(A) = 35 and n(A') = 25, then find the value of n(U).
n(U) = n(A) + n(A') = 35 + 25 = 60
d) Out of two sets P and Q, there are 40 elements in P, 60 elements in (P∪Q) and 10 elements in (P∩Q). How many elements are there in Q?
n(P∪Q) = n(P) + n(Q) - n(P∩Q)
60 = 40 + n(Q) - 10
60 = 30 + n(Q)
n(Q) = 30
4. Survey Problems
a) In a survey of 180 students of a school:
Given: Nepali only = 45, English only = 60, None = 15, Total = 180
i) Show the above information in a Venn-diagram.
ii) Find the number of students who like both the subjects.
Let x = number who like both subjects
Total students = Nepali only + English only + Both + None
180 = 45 + 60 + x + 15
180 = 120 + x
x = 60
So, 60 students like both subjects.
iii) Find the number of students who like at least one subject.
Students who like at least one subject = Total - None
= 180 - 15 = 165
Or: Nepali only + English only + Both = 45 + 60 + 60 = 165
b) In a survey among the 1200 students of a school:
Given: Math only = 100, Science only = 200, Neither = 700, Total = 1200
i) Show the above information in a Venn-diagram.
ii) Find the number of students who like both the subjects.
Let x = number who like both subjects
Total students = Math only + Science only + Both + Neither
1200 = 100 + 200 + x + 700
1200 = 1000 + x
x = 200
So, 200 students like both subjects.
c) In a survey among 60 students:
Given: Football only = 10, Volleyball only = 20, Neither = 12, Total = 60
i) Show the above information in a Venn-diagram.
ii) Find the number of students who play both the games.
Let x = number who play both games
Total students = Football only + Volleyball only + Both + Neither
60 = 10 + 20 + x + 12
60 = 42 + x
x = 18
So, 18 students play both games.
iii) Find the number of students who play at least one game.
Students who play at least one game = Total - Neither
= 60 - 12 = 48
Or: Football only + Volleyball only + Both = 10 + 20 + 18 = 48
5. Newspaper and Music Survey Problems
a) Survey among 900 people:
Given: Madhupark readers = 525, Yubamanch readers = 450, Neither = 75, Total = 900
i) Show the information in a Venn-diagram.
ii) Find the number of people who read both the newspapers.
Let x = number who read both newspapers
Madhupark only = 525 - x
Yubamanch only = 450 - x
Total = Madhupark only + Yubamanch only + Both + Neither
900 = (525 - x) + (450 - x) + x + 75
900 = 525 + 450 + 75 - x
900 = 1050 - x
x = 1050 - 900 = 150
So, 150 people read both newspapers.
iii) Find the number of people who read only one newspaper.
Madhupark only = 525 - 150 = 375
Yubamanch only = 450 - 150 = 300
Only one newspaper = 375 + 300 = 675
b) Survey among 150 people:
Given: Modern songs = 90, Folk songs = 70, Neither = 30, Total = 150
i) Show the information in a Venn-diagram.
ii) Find the number of people who like both the songs.
Let x = number who like both songs
Modern only = 90 - x
Folk only = 70 - x
Total = Modern only + Folk only + Both + Neither
150 = (90 - x) + (70 - x) + x + 30
150 = 90 + 70 + 30 - x
150 = 190 - x
x = 190 - 150 = 40
So, 40 people like both songs.
iii) Find the number of people who like only modern songs.
Only modern songs = 90 - 40 = 50
c) Survey among 360 players:
Given: Volleyball = 210, Football = 180, Neither = 30, Total = 360
i) Show the information in a Venn-diagram.
ii) Find the number of players who like to play both the games.
Let x = number who like both games
Volleyball only = 210 - x
Football only = 180 - x
Total = Volleyball only + Football only + Both + Neither
360 = (210 - x) + (180 - x) + x + 30
360 = 210 + 180 + 30 - x
360 = 420 - x
x = 420 - 360 = 60
So, 60 players like both games.
iii) Find the number of people who like to play only one game.
Volleyball only = 210 - 60 = 150
Football only = 180 - 60 = 120
Only one game = 150 + 120 = 270
6. Percentage Problems
a) Examination results:
Given: Pass English = 70%, Pass Math = 60%, Fail both = 20%, Pass both = 550 students
i) Show the above information in a Venn-diagram.
ii) Find the total number of students participated in the examination.
Let total students = T
Pass English = 0.7T
Pass Math = 0.6T
Fail both = 0.2T
Pass both = 550
Using the formula: n(E∪M) = n(E) + n(M) - n(E∩M)
Also, n(E∪M) = T - n(E∪M)' = T - 0.2T = 0.8T
So, 0.8T = 0.7T + 0.6T - 550
0.8T = 1.3T - 550
1.3T - 0.8T = 550
0.5T = 550
T = 550 ÷ 0.5 = 1100
So, total students = 1100
iii) Find how many students passed English only.
Pass English only = Pass English - Pass both
= 0.7T - 550
= 0.7 × 1100 - 550
= 770 - 550 = 220
So, 220 students passed English only.
b) According to a survey of students:
Given: Science = 60%, Management = 70%, Neither = 10%, Both = 400 students
i) Show the above information in a Venn-diagram.
ii) Find how many students were participated in the survey.
Let total students = T
Science only = 60% - Both%
Management only = 70% - Both%
Neither = 10%
Total percentage = Science only + Management only + Both + Neither = 100%
Let Both% = x%
(60 - x) + (70 - x) + x + 10 = 100
140 - x = 100
x = 40%
So, 40% of students are interested in both = 400 students
Therefore, 40% of T = 400
T = 400 ÷ 0.4 = 1000
So, total students = 1000
iii) Find the number of students who are interested to study Science only.
Science only = 60% - Both% = 60% - 40% = 20%
Number of students = 20% of 1000 = 0.2 × 1000 = 200
So, 200 students are interested in Science only.
c) In a survey among people of a community:
Given: Motorcycle = 65%, Scooter = 35%, Both = 20%, Both (actual) = 200 people
i) Show the above information in a Venn-diagram.
ii) Find how many people were participated in the survey.
Let total people = T
Both = 20% of T = 200
So, 0.2T = 200
T = 200 ÷ 0.2 = 1000
So, total people = 1000
iii) Find the number of people who ride motorcycles only.
Motorcycle only = Total motorcycle - Both
= 65% - 20% = 45%
Number of people = 45% of 1000 = 0.45 × 1000 = 450
So, 450 people ride motorcycles only.
7. Ratio Problems
a) Among 95 people of a community:
Given: Total = 95, Tea:Coffee = 4:5, Tea only = 10, Neither = 15
i) Show the information in a Venn-diagram.
ii) Find the number of people who drinks exactly one of tea or coffee.
Let the ratio multiplier be x
Tea drinkers = 4x, Coffee drinkers = 5x
Let Both = y
From the Venn diagram:
Tea only = Tea drinkers - Both = 4x - y = 10
Coffee only = Coffee drinkers - Both = 5x - y
Neither = 15
Total = Tea only + Coffee only + Both + Neither = 95
10 + (5x - y) + y + 15 = 95
25 + 5x = 95
5x = 70
x = 14
Now, Tea only = 4x - y = 10
4×14 - y = 10
56 - y = 10
y = 46
Coffee only = 5x - y = 5×14 - 46 = 70 - 46 = 24
Exactly one drink = Tea only + Coffee only = 10 + 24 = 34
So, 34 people drink exactly one of tea or coffee.
iii) Find the number of people who drinks at least one; either tea or coffee.
At least one = Total - Neither = 95 - 15 = 80
Or: Tea only + Coffee only + Both = 10 + 24 + 46 = 80
So, 80 people drink at least one of tea or coffee.
b) In a survey of 64 students of a class:
Given: Total = 64, Milk only : Curd only = 2:1, Both = 16
i) Show the above information in a Venn-diagram.
ii) Find the number of students who like milk.
Let Milk only = 2x, Curd only = x
Both = 16
Neither = Total - (Milk only + Curd only + Both)
Neither = 64 - (2x + x + 16) = 64 - (3x + 16) = 48 - 3x
Since Neither cannot be negative, 48 - 3x ≥ 0 ⇒ x ≤ 16
Also, we need more information to find exact value of x
Assuming there's no additional constraint and using the ratio only:
Total milk likers = Milk only + Both = 2x + 16
We need the value of x to find the exact number
Note: The question seems to have insufficient information to determine the exact number without additional constraints.
iii) Find the number of students who like only one kind of drink.
Only one kind = Milk only + Curd only = 2x + x = 3x
Again, we need the value of x to find the exact number
Note: With the given information, we can only express the answer in terms of x.
7. Conference Survey Problem
c) In a conference of 320 participants:
Given: Total = 320, Sing only = 60, Dance only = 100, Neither = 3 × Both
i) Show the above information in a Venn-diagram.
ii) Find how many people do not do both genres.
Let Both = x
Neither = 3x
Total = Sing only + Dance only + Both + Neither
320 = 60 + 100 + x + 3x
320 = 160 + 4x
4x = 320 - 160 = 160
x = 40
Neither = 3x = 3 × 40 = 120
So, 120 people do not do both genres.
iii) Find the number of people who do one genre at most.
"One genre at most" means they do either one genre or none
This includes: Sing only + Dance only + Neither
= 60 + 100 + 120 = 280
So, 280 people do one genre at most.
8. Gadget Usage Survey
According to a survey of 200 people:
Given: Total = 200, Laptop only : Mobile only = 2:3, Both = 30%, Neither = 15%
i) Show the above information in a Venn-diagram.
ii) Find the number of people who uses laptop.
Total percentage = 100%
Laptop only + Mobile only + Both + Neither = 100%
Let Laptop only = 2x%, Mobile only = 3x%
2x + 3x + 30 + 15 = 100
5x + 45 = 100
5x = 55
x = 11
Laptop only = 2x% = 22%
Total laptop users = Laptop only + Both = 22% + 30% = 52%
Number of laptop users = 52% of 200 = 0.52 × 200 = 104
So, 104 people use laptop.
iii) Find how many people use one gadget at most.
"One gadget at most" means they use either one gadget or none
This includes: Laptop only + Mobile only + Neither
= 22% + 33% + 15% = 70%
Number of people = 70% of 200 = 0.7 × 200 = 140
So, 140 people use one gadget at most.
9. Sports Players Survey
Out of 300 players in a survey:
Given: Total = 300, Volleyball only = 1/3 of total, Football only = 60% of remaining, Neither = 60
Find the ratio of the number of players who play volleyball and football.
Volleyball only = 1/3 × 300 = 100
Remaining players = 300 - 100 = 200
Football only = 60% of 200 = 0.6 × 200 = 120
Neither = 60
Now, Both = Total - (Volleyball only + Football only + Neither)
= 300 - (100 + 120 + 60) = 300 - 280 = 20
Total volleyball players = Volleyball only + Both = 100 + 20 = 120
Total football players = Football only + Both = 120 + 20 = 140
Ratio of volleyball to football players = 120:140 = 6:7
So, the ratio is 6:7
Venn-diagram representation:
10. Volleyball and Cricket Survey
Among 65 players participated in a survey:
Given: Total = 65, Volleyball only = 11, Cricket only = 33, Cricket players = 2 × Volleyball players
Find the number of players who play both and who do not play both.
Let Both = x
Total volleyball players = Volleyball only + Both = 11 + x
Total cricket players = Cricket only + Both = 33 + x
Given: Cricket players = 2 × Volleyball players
33 + x = 2(11 + x)
33 + x = 22 + 2x
33 - 22 = 2x - x
11 = x
So, Both = 11
Neither = Total - (Volleyball only + Cricket only + Both)
= 65 - (11 + 33 + 11) = 65 - 55 = 10
So, 11 players play both games, and 10 players do not play both.
Venn-diagram representation:
11. Fruit Preference Survey
In a survey of 80 people:
Given: Total = 80, Orange = 60, Both = 10, Orange = 5 × Apple
Find the number of people who like apples only and those who do not like both fruits.
Orange = 60, Both = 10
Orange only = Orange - Both = 60 - 10 = 50
Given: Orange = 5 × Apple
60 = 5 × Apple
Apple = 60 ÷ 5 = 12
Apple only = Apple - Both = 12 - 10 = 2
Neither = Total - (Orange only + Apple only + Both)
= 80 - (50 + 2 + 10) = 80 - 62 = 18
So, 2 people like apples only, and 18 people do not like both fruits.
Venn-diagram representation:
Project Work
Instructions:
1. Form groups of five students each
2. Go to different classes in your school
3. Ask the question: "Which of the following game do you like?"
(a) Cricket (b) Football (c) Cricket and football both (d) Others
4. Collect the data and present it in a Venn-diagram
5. Discuss your findings in the class
Sample Venn-diagram for project work:
Key Formulas Used:
n(A∪B) = n(A) + n(B) - n(A∩B)
n(A') = n(U) - n(A)
n(A only) = n(A) - n(A∩B)
n(Neither) = n(U) - n(A∪B)
For percentage problems: Convert percentages to actual numbers using total
For ratio problems: Use ratio multiplier to find exact values
Important Notes:
1. "At most one" means either one or none (excludes both)
2. Always verify your answers by checking if all parts add up to the total
3. Pay attention to the wording: "only", "both", "neither", "at most one"
4. For ratio problems, use a common multiplier to find exact values
5. Some questions may have insufficient information and require additional constraints
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