Chapter-1 (Sets) Exercise Solutions

Sets: Cardinality of Sets

Mathematics • Grade 10 • Unit 1 — Notes, Formulas, Solved Examples & Practice

In this post

1. Cardinality of Two Sets
2. Solved Examples (Two Sets)
3. Cardinality of Three Sets
4. Solved Examples (Three Sets)
5. Practice Questions with Answers

1.1 Cardinality of Two Sets

The cardinality of a set, written n(A), is simply the number of elements in it. A Venn diagram helps us split a class into groups such as “likes only coffee”, “likes both” and “likes neither”.

Warm-up: coffee and tea

In a class: 15 like coffee, 10 like tea, 6 like both and 5 like neither.

• Coffee only = 15 − 6 = 9
• Tea only = 10 − 6 = 4
• Total students = 9 + 6 + 4 + 5 = 24

Figure: coffee (C) and tea (T)

Key formulas — if A and B are overlapping sets

1. n(A∪B) = n(A) + n(B) − n(A∩B)
2. Only A: n_o(A) = n(A) − n(A∩B)
3. Only B: n_o(B) = n(B) − n(A∩B)
4. n(A∪B) = n_o(A) + n_o(B) + n(A∩B)
5. If A and B are disjoint: n(A∪B) = n(A) + n(B)
6. If U contains only elements of A and B: n(U) = n(A∪B)
7. Otherwise: n(U) = n(A∪B) + n(A∪B)′  (the second term = “neither”)

Phrase in the question	Meaning / formula
At least one	n(A∪B) = n(A) + n(B) − n(A∩B)
At most one	n(U) − n(A∩B)
Exactly one / only one	no(A) + no(B) = n(A) + n(B) − 2·n(A∩B)
Neither	n(A∪B)′ = n(U) − n(A∪B)

Solved Examples — Two Sets

Example 1 In a survey of 300 people, 175 like cricket, 150 like football and 25 like neither. Find (a) how many like both, (b) how many like exactly one game.

Let n(C∩F) = x. Then only cricket = 175 − x, only football = 150 − x.

(175 − x) + x + (150 − x) + 25 = 300 350 − x = 300 ⇒ x = 50

(a) 50 people like both games

(b) Only cricket = 125, only football = 100 → 225 like exactly one game

Example 2 Among 120 grade 10 students, 30 like only Mathematics, 40 like only English and 10 like neither. Find (a) both, (b) at least one.

Let n(M∩E) = x.

30 + x + 40 + 10 = 120 ⇒ x = 40

Both: 40   At least one: 30 + 40 + 40 = 110

Example 3 In a survey of SEE students, 75% want to study science, 55% want to study staff nurse, 5% gave no answer, and 21 students want both. Find (a) the total surveyed, (b) how many want only staff nurse.

Let n(U) = x. Only S = 0.75x − 21, only N = 0.55x − 21, neither = 0.05x.

(0.75x − 21) + 21 + (0.55x − 21) + 0.05x = x 1.35x − 21 = x ⇒ 0.35x = 21 ⇒ x = 60

Total surveyed: 60

Only staff nurse = 0.55 × 60 − 21 = 33 − 21 = 12

Example 4 Of 300 foreign tourists, the ratio of those who visited Pokhara to those who visited Lumbini is 2 : 3. 90 visited both and 60 visited neither. Find (a) tourists who visited only one place, (b) at least one place.

Let n(P) = 2x and n(L) = 3x. Only P = 2x − 90, only L = 3x − 90.

(2x − 90) + 90 + (3x − 90) + 60 = 300 ⇒ 5x = 330 ⇒ x = 66

Only P = 2(66) − 90 = 42; only L = 3(66) − 90 = 108.

Only one place: 42 + 108 = 150   At least one: 300 − 60 = 240

Example 5 Among 200 students, the ratio of those who like only Mathematics to only English is 2 : 3. 30% like both and 15% like neither. Find the difference between the number who like Mathematics and the number who like English.

Both = 30% of 200 = 60; neither = 15% of 200 = 30. Let only M = 2x, only E = 3x.

2x + 60 + 3x + 30 = 200 ⇒ 5x = 110 ⇒ x = 22

n(M) = 2(22) + 60 = 104,   n(E) = 3(22) + 60 = 126

Difference = 126 − 104 = 22

1.2 Cardinality of Three Sets

With three overlapping sets A, B and C, a Venn diagram has seven regions inside the circles plus the area outside them. Always fill the diagram in this order: (1) all three → (2) exactly two → (3) exactly one → (4) none.

Figure: the eight regions of a three-set Venn diagram

Key formulas — A, B, C overlapping

1. n_o(A∩B) = n(A∩B) − n(A∩B∩C)
2. n_o(B∩C) = n(B∩C) − n(A∩B∩C)
3. n_o(A∩C) = n(A∩C) − n(A∩B∩C)
4. n(A∪B∪C) = n(A) + n(B) + n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C)
5. n(U) = n(A∪B∪C) + n(A∪B∪C)′
6. If A, B, C are disjoint: n(A∪B∪C) = n(A) + n(B) + n(C)

Activity In a class, 40 like orange, 35 like mango, 50 like banana; 15 like orange & mango, 20 like mango & banana, 25 like orange & banana; 5 like all three; 30 like none. How many students were surveyed?

Region	Working	Value
All three	given	5
O & M only	15 − 5	10
M & B only	20 − 5	15
O & B only	25 − 5	20
O only	40 − (5 + 10 + 20)	5
M only	35 − (5 + 10 + 15)	5
B only	50 − (5 + 15 + 20)	10
None	given	30

Total = 5 + 10 + 15 + 20 + 5 + 5 + 10 + 30 = 100 students

Solved Examples — Three Sets

Example 1 n(U) = 120, n(A) = 48, n(B) = 51, n(C) = 40, n(A∩B) = 11, n(B∩C) = 10, n(A∩C) = 9, n(A∩B∩C) = 4. Find n(A∪B∪C) and n(A∪B∪C)′.

n(A∪B∪C) = 48 + 51 + 40 − 11 − 10 − 9 + 4 = 113 n(A∪B∪C)′ = 120 − 113 = 7

113 and 7

Example 2 Of 180 students, 86 passed Science, 80 passed Maths, 76 passed Nepali; 26 passed Science & Maths, 36 Maths & Nepali, 32 Science & Nepali; 20 passed none. Find how many passed all three.

180 = 86 + 80 + 76 − 26 − 36 − 32 + n(S∩M∩N) + 20 180 = 168 + n(S∩M∩N) ⇒ n(S∩M∩N) = 12

12 students passed all three subjects

Alternative: let the “all three” region be x, write every region in terms of x, add all regions plus 20 and set the sum equal to 180. You get 168 + x = 180, so x = 12.

Example 3 36 students got medals in dance, 12 in drama and 18 in music. Only 45 students won any medal and 4 won in all three events. How many won medals in exactly two events?

From the union formula: 45 = 36 + 12 + 18 − [n(A∩B) + n(B∩C) + n(A∩C)] + 4

n(A∩B) + n(B∩C) + n(A∩C) = 70 − 45 = 25

Each pairwise overlap includes the 4 who won all three, so subtract 4 three times:

Exactly two = 25 − 3(4) = 13

13 students

Practice Questions

Try each one on paper first, then tap to check your answer.

Q1. Of 180 students, 45 like Nepali only, 60 like English only and 15 like neither. Find how many like both, and how many like at least one subject.

Both = 60; at least one = 165

Q2. Among 150 people, 90 like modern songs, 70 like folk songs and 30 like neither. Find how many like both and how many like only modern songs.

Both = 40; only modern = 50

Q3. In a survey of 1200 students, 100 like Mathematics only, 200 like Science only and 700 like neither. How many like both?

Both = 200  (1200 − 700 − 100 − 200 = 200)

Q4. If n(A) = 35 and n(A′) = 25, find n(U).

n(U) = 60

Q5. Of 90 students, 43 passed Science, 40 Maths, 38 Nepali; 13 passed Science & Maths, 18 Maths & Nepali, 16 Science & Nepali; 10 passed none. How many passed all three?

80 passed at least one. 80 = 43 + 40 + 38 − 13 − 18 − 16 + x = 74 + x. x = 6

Q6. In an exam, 40% passed Science, 45% Maths, 50% Nepali; 10% Science & Maths, 20% Maths & Nepali, 15% Science & Nepali; 5% failed all. Find the percentage who passed all three, and at least one.

At least one = 100 − 5 = 95%. 95 = 135 − 45 + x ⇒ all three = 5%; at least one = 95%

Q7. Among 65 players, 11 play volleyball only and 33 play cricket only. The number who play cricket is double the number who play volleyball. Find how many play both and how many play neither.

Let both = x: 33 + x = 2(11 + x) ⇒ x = 11?? Check: 33 + x = 22 + 2x ⇒ x = 11. Neither = 65 − (11 + 11 + 33) = 10. Both = 11; neither = 10

Project idea: Ask students in different classes which game they like (cricket, football, both, or others). Record the results, draw a Venn diagram and discuss what you find in class.

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