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Mathematics questions, chapter 11
Nama / NameKelas / ClassTarikh / Date

Introduction of set

40 marks. Use Practice to check each answer as you go, or Exam to answer everything first. Do not use a calculator. Show your working for Section B.

Section A: objective questions

12 marks
Choose the correct answer.
1

Question 1

Which symbol represents the universal set?
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B
ξ (xi) is the universal set. φ is the empty set, ∈ means "is an element of" and ⊂ means "is a subset of".
For the teacher: Remembering, low difficulty, standard 11.2.1
Textbook p. 254
2

Question 2

P = {letters in the word 'KELANTAN'}. Which of the following is set P written by listing?
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A
Each letter is written once only, so P = {K, E, L, A, N, T}. Writing A and N twice is wrong. {E, A} is only the vowels and {K, L, N, T} only the consonants.
For the teacher: Understanding, low difficulty, standard 11.1.2
Textbook p. 249
3

Question 3

Which of the following is an empty set?
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C
No month has 32 days, so this set has no elements. {0} has one element (0), {even prime numbers} = {2} and {multiples of 5 less than 6} = {5}.
For the teacher: Understanding, low difficulty, standard 11.1.2
Textbook p. 250
4

Question 4

Given P = {x : x is a factor of 24}, which statement is true?
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D
P = {1, 2, 3, 4, 6, 8, 12, 24}, so 8 ∈ P. 5 is not a factor of 24. 48 is a multiple of 24, not a factor. 6 is a factor, so 6 ∈ P, not 6 ∉ P.
For the teacher: Applying, low difficulty, standard 11.1.3
Textbook p. 251
5

Question 5

Q = {x : x is a two-digit number such that the sum of its digits is 6}. Find n(Q).
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B
Q = {15, 24, 33, 42, 51, 60}, so n(Q) = 6. 5 comes from forgetting 60 (6 + 0 = 6). 06 is not a two-digit number.
For the teacher: Applying, medium difficulty, standard 11.1.4
Textbook p. 252
6

Question 6

Which pair of sets are equal sets?
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A
The factors of 6 are 1, 2, 3 and 6, exactly the same elements. {0} has one element but φ has none. {1, 2, 3} and {a, b, c} have the same number of elements but different elements. 12 is not less than 10.
For the teacher: Understanding, medium difficulty, standard 11.1.5
Textbook p. 252
7

Question 7

Given ξ = {x : x is an integer and 1 ≤ x ≤ 12} and P = {multiples of 3}, find P′.
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D
P = {3, 6, 9, 12}. P′ is every other element of ξ: {1, 2, 4, 5, 7, 8, 10, 11}. Check: 4 + 8 = 12 = n(ξ). {3, 6, 9, 12} is P itself; the other two options leave out 1 or 11.
For the teacher: Applying, medium difficulty, standard 11.2.1
Textbook p. 255
8

Question 8

How many subsets does the set {prime numbers which are less than 10} have?
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C
The set is {2, 3, 5, 7}, with 4 elements, so it has 24 = 16 subsets. 15 forgets the empty set φ, and 4 is only the number of elements.
For the teacher: Applying, medium difficulty, standard 11.2.3
Textbook p. 258
9

Question 9

Given A = {2, 4, 6, 8}, which of the following is not a subset of A?
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D
5 ∉ A, so {4, 5} ⊄ A. The empty set is a subset of every set, and A is a subset of itself.
For the teacher: Understanding, medium difficulty, standard 11.2.3
Textbook p. 257
10

Question 10

Diagram 1 is a Venn diagram showing the universal set ξ and set M. List the elements of M′.
Venn diagram: M contains 21, 24, 27, 30; 20, 22, 23, 25, 26, 28, 29 are outside M
Diagram 1
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B
M′ is the elements inside the rectangle but outside M: {20, 22, 23, 25, 26, 28, 29}. {21, 24, 27, 30} is M, the third option is ξ, and the last option leaves out 20.
For the teacher: Analysing, medium difficulty, standard 11.2.2
Textbook p. 256
11

Question 11

Diagram 2 is a Venn diagram showing the universal set ξ and sets A, B and C. Which statement is true?
Rectangle xi; oval B drawn inside oval A; triangle C drawn outside A
Diagram 2
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A
B is drawn completely inside A, so B ⊂ A. C is outside A, so C ⊄ A. A ⊂ B is the wrong way round: the bigger set cannot fit inside the smaller one.
For the teacher: Analysing, high difficulty, standard 11.2.5
Textbook p. 259
12

Question 12

Diagram 3 is a Venn diagram showing the universal set ξ and sets P and Q. Find n(Q′).
Venn diagram: Q = {b, d, f} inside P; a, c, e in P only; g, h, i, j outside P
Diagram 3
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C
Q′ is everything in ξ outside Q: a, c, e (inside P but outside Q) and g, h, i, j. So n(Q′) = 3 + 4 = 7. 4 counts only the elements outside P, and 3 is n(Q).
For the teacher: Analysing, high difficulty, standard 11.2.5
Textbook p. 260

Section B: structured questions

28 marks
Answer all questions. Show your working.
13

Question 13

S is the set of multiples of 4 which are less than 30.
[1 mark] (a)Write set S by listing.
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Tick each point you got.
[1 mark] (b)Write set S using set builder notation.
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Tick each point you got.
[1 mark] (c)Describe the set T = {a, e, i, o, u} by using description.
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Tick each point you got.
For the teacher: Understanding, low difficulty, standard 11.1.2
Textbook p. 249
14

Question 14

It is given that P = {x : x is an odd number and 0 < x < 20} and Q = {x : x is a common multiple of 3 and 4, 0 < x < 50}.
[1 mark] (a)15 ___ P
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∈
15 is odd and between 0 and 20.
[1 mark] (b)24 ___ Q
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∈
24 = 3 × 8 = 4 × 6, so 24 is a common multiple of 3 and 4.
[1 mark] (c)List the elements of Q and find n(Q).
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Tick each point you got.
For the teacher: Applying, low difficulty, standard 11.1.3
Textbook p. 251
15

Question 15

Given that P = {2, 5, y, 11, 3, 13, 17} and Q = {prime numbers which are less than 18}. P and Q are equal sets.
[1 mark] (a)List the elements of Q.
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Tick each point you got.
[1 mark] (b)Find the value of y. Give a reason.
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Tick each point you got.
[1 mark] (c)Find n(P).
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Tick each point you got.
For the teacher: Analysing, medium difficulty, standard 11.1.5
Textbook p. 253
16

Question 16

Given that ξ = {x : x is an integer and 10 ≤ x ≤ 20} and A = {multiples of 4}.
[2 marks] (a)List the elements of ξ and of A.
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Tick each point you got.
[1 mark] (b)Find n(A′).
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Check: n(A) + n(A′) = 3 + 8 = 11 = n(ξ).
Tick each point you got.
[1 mark] (c)Draw a Venn diagram to represent ξ and A. Shade the region that represents A′.
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Venn diagram: A contains 12, 16, 20; the region outside A is shaded and contains 10, 11, 13, 14, 15, 17, 18, 19
Tick each point you got.
For the teacher: Applying, medium difficulty, standard 11.2.2
Textbook p. 256
17

Question 17

A burger stall offers three extra toppings: cheese, egg and chicken. A customer may choose none, some or all of the toppings. Each possible choice is a subset of K = {cheese, egg, chicken}.
[2 marks] (a)List all the possible choices a customer can make.
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Tick each point you got.
[1 mark] (b)The stall adds mushroom as a fourth topping. How many possible choices are there now?
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Tick each point you got.
For the teacher: Applying, medium difficulty, standard 11.2.3
Textbook p. 262
18

Question 18

Diagram 4 is a Venn diagram showing the universal set ξ and sets P, Q and R.
Venn diagram: xi = 1 to 15. Q = {1, 2, 4} inside P; P also contains 3, 6, 12; R = {5, 10, 15} separate from P; 7, 8, 9, 11, 13, 14 outside all sets
Diagram 4
[1 mark] (a)State the relationship between set Q and set P using a symbol.
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Tick each point you got.
[1 mark] (b)List the elements of P′.
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Tick each point you got.
[1 mark] (c)Find n(R′).
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Tick each point you got.
[1 mark] (d)R ⊂ P′
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True
5, 10 and 15 are all outside P, so every element of R is in P′.
For the teacher: Analysing, high difficulty, standard 11.2.5
Textbook p. 259
19

Question 19

Diagram 5 shows Hafiz's working to find n(A′).
Hafiz's working: xi = {1, 2, 3, ..., 15}, A = {factors of 12}. Line 1: A = {2, 3, 4, 6}. Line 2: A' = {1, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15}. Line 3: n(A') = 11
Diagram 5
[2 marks] (a)In which line did Hafiz make his first mistake? Explain the mistake.
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Tick each point you got.
[2 marks] (b)Write the correct A′ and find n(A′).
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Check: n(A) + n(A′) = 6 + 9 = 15 = n(ξ).
Tick each point you got.
For the teacher: Evaluating, high difficulty, standard 11.2.1
Textbook p. 255
20

Question 20

The universal set is ξ = {1, 2, 3, …, 12}. Design two sets A and B such that n(A) = 6, n(B) = 3, B ⊂ A and 7 ∈ A′.
[2 marks] (a)Describe your sets A and B, then write them by listing.
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Tick each point you got.
[2 marks] (b)Draw a Venn diagram to represent ξ, A and B.
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Sample answer (other correct sets are accepted):
Venn diagram: B = {4, 8, 12} inside A, which also has 2, 6, 10; 1, 3, 5, 7, 9, 11 outside A
Tick each point you got.
For the teacher: Creating, high difficulty, standard 11.2.5
Textbook p. 259