Unit 1 – Sets | Nepal eNotes

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MATHEMATICS • UNIT 1

Unit 1: Sets

Exercise 1.1 & Exercise 1.2 — supplied 39-page handwritten solution converted into WordPress-ready HTML.

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Important Set Formulae

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

n(A ∪ B ∪ C) = n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(C∩A) + n(A∩B∩C)

n(A′) = n(U) − n(A)

n((A ∪ B)′) = n(U) − n(A ∪ B)

n(A Δ B) = n(A) + n(B) − 2n(A ∩ B)

n(A only) = n(A) − n(A ∩ B)

PDF Pages 1–29

Exercise 1.1

PDF Page 1

1. Basic Cardinality Questions

(b) For two sets A and B, if A ⊂ B, find n(A ∪ B) and n(A ∩ B).

A ∪ B = B

∴ n(A ∪ B) = n(B)

A ∩ B = A

∴ n(A ∩ B) = n(A)

(c) If A and B are overlapping sets, state the formula for n(A ∪ B).

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

(d) There are 12 and 8 elements in sets A and B respectively. Find the minimum number of elements in A ∪ B.

Given n(A)=12, n(B)=8.

The minimum occurs when the smaller set is contained in the larger set.

∴ minimum n(A ∪ B)=12.

PDF Page 2

2. Cardinality from a Two-Set Venn Diagram

The source diagram contains M only = 20, M ∩ E = 60, E only = 30 and 15 outside both sets.

Required	Working	Answer
n(M)	20+60	80
n(E)	60+30	90
n(M∪E)	20+60+30	110
n(M∩E)	intersection	60
Neither	outside both circles	15
n(U)	20+60+30+15	125

PDF Pages 2–5

3. Cardinality Problems

(a) n(U)=200, n(M)=2m, n(E)=3m, n(M∩E)=60 and outside both=40.

The handwritten source uses:

200 = 2m + 3m + 60 + 40

200 = 5m + 100

∴ m=20

Source fidelity: This working is reproduced as written in the source and is not silently corrected.

(b) n(U)=350, n(A)=200, n(B)=220, n(A∩B)=120. Find n((A∪B)′).

n(A∪B)=200+220−120=300

n((A∪B)′)=350−300=50

(c) n(A)=35 and n(A′)=25. Find n(U).

n(U)=35+25=60

(d) n(P)=40, n(P∪Q)=60, n(P∩Q)=10. Find n(Q).

60=40+n(Q)−10

∴ n(Q)=30

PDF Pages 5–13

4–5. Survey Problems Using Two Sets

Nepali and English Survey

The source Venn regions are 45, n, 60 and 15.

180=45+n+60+15

∴ n=60

At least one subject =45+60+60=165

Mathematics and Science Survey

1200=100+n+200+700

∴ n=200 like both.

At least one=100+200+200=500

Football and Volleyball Survey

60=10+n+20+12

∴ n=18 play both.

At least one=10+20+18=48

Two Newspapers

900=525+450−n+75

∴ n=150 read both.

Only one=(525−150)+(450−150)=675

Modern and Folk Songs

150=90+70−n+30

∴ n=40 like both.

Only modern=90−40=50

Volleyball and Basketball

360=210+180−n+30

∴ n=60 like both.

Only volleyball=150, only basketball=120.

Only one game=270

PDF Pages 14–18

6. Percentage-Based Survey Problems

(a) English and Mathematics Examination

70% passed English, 60% passed Mathematics, 20% failed both and 550 passed both.

n=0.7n−550+550+0.6n−550+0.2n

n=1.5n−550

∴ n=1100

English only=0.7×1100−550=220

(b) Science and Management

60% are interested in Science, 70% in Management, 10% in neither and 400 in both.

n=0.6n−400+400+0.7n−400+0.1n

∴ n=1000

Science only=600−400=200

(c) Motorcycle and Scooter

65% ride a motorcycle, 35% a scooter, 20% both and 200 neither.

n=0.65n−0.2n+0.2n+0.35n−0.2n+200

∴ n=1000

Motorcycle only=650−200=450

PDF Pages 19–29

Additional Two-Set Problems

Tea and Coffee

The source obtains m=10.

Exactly one of tea or coffee=70.

At least one=80.

Source note: Some values in pages 19–20 are difficult to read consistently, so the final handwritten results are retained.

Milk and Curd

Let n(M)=2n and n(C)=n; both=16.

64=3n+16 ⇒ n=16.

Milk=2n=32.

The handwritten source gives one kind of drink=48.

Conference: Singing and Dancing

The handwritten solution obtains n=40.

Outside the activities=3n=120.

At most one activity=320−40=280.

Laptop and Mobile

Among 200 people, laptop-only : mobile-only = 2:3, 30% use both and 15% use neither.

Both=60, neither=30.

200=2m+3m+60+30 ⇒ 5m=110 ⇒ m=22.

Laptop users=44+60=104.

One gadget at most=200−60=140.

Volleyball and Football

Volleyball only=1/3×300=100.

Football only=60% of 200=120.

The source obtains intersection=20.

Total volleyball=120; total football=140; ratio=6:7.

Volleyball and Cricket

Cricket only=33, volleyball only=11.

33+n=2(11+n) ⇒ n=11 play both.

65=11+11+33+y ⇒ y=10 play neither.

Orange and Apple

The final handwritten results on pages 28–29 give apple-only=2 and outside both=18.

PDF Pages 30–39

Exercise 1.2

PDF Page 30

1. Cardinalities from a Three-Set Venn Diagram

The source asks for several cardinalities from a given P–Q–R Venn diagram. The handwritten answer line gives:

Part	Answer in source
(a)	4
(b)	6
(c)	14
(d)	9
(e)	3
(f)	2
(g)	12
(h)	2
(i)	2

The set-expression labels in the photographed diagram are faint, so these numerical results are reproduced without re-labelling ambiguous expressions.

PDF Pages 30–31

2. Multiples of 2, 3 and 5 Below 30

U is the set of positive integers less than 30.

P={2,4,6,8,10,12,14,16,18,20,22,24,26,28}

Q={3,6,9,12,15,18,21,24,27}

R={5,10,15,20,25}

(a) P and Q

n(P∪Q)=n(P)+n(Q)−n(P∩Q)

=14+9−4=19

(b) Three-set union

Using inclusion–exclusion, the handwritten solution obtains 21.

PDF Pages 31–33

3. Three-Set Cardinality Formula Problems

n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(C∩A)+n(A∩B∩C)

Pages 31–33 apply this formula to several examples, including universal-set values such as 100, 105 and 120.

Some pairwise-intersection symbols and numerals are visually ambiguous in the scan, so unclear values have not been silently replaced by inferred ones.

PDF Pages 33–34

4. Examination Result Survey

Out of 90 students, the source gives pass information for Science, Mathematics and Nepali and represents the values in a three-set Venn diagram.

The following page sums the Venn regions against the total 90 to obtain the remaining region.

PDF Pages 34–36

5. Volleyball, Basketball and Cricket Survey

n(V)=23
n(B)=15
n(C)=20
n(V∩B)=7
n(B∩C)=5
n(C∩V)=8
15 played none

The handwritten Venn solution obtains:

All three games=3

Only volleyball=15

Only cricket=14

PDF Pages 36–38

6. Percentage-Based Three-Subject Survey

The source uses 1000 students with:

Science=40%
Mathematics=25%
Nepali=50%
Science∩Mathematics=10%
Mathematics∩Nepali=20%
Science∩Nepali=15%
All three=5%

n(S∪M∪N)=40+25+50−10−20−15+5=75%

Outside all three=100−75=25%

Pages 37–38 continue with questions on only one subject, exactly two subjects and at least one subject.

PDF Pages 38–39

Final Three-Set Venn-Diagram Exercise

The final handwritten exercise gives:

Total participants=64
Exactly two sets=17
The number in at least one set is found by adding the regions inside the three circles.

A few region labels in the final diagram are faint. Clearly readable source results are preserved without inventing missing values.