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1 © Penerbitan Pelangi Sdn. Bhd.
Paper 1
1. { }
Answer: A
2. P = {2, 4, 6, 8}Answer: C
3. d Q (d is an element of set Q)
Answer: D
4.
b
a
e
dP
Answer: A
5. The number of elements in set R, n(R) = 3
Answer: B
6. The elements of set H = {a, b, c, d, e} are a, b, c, d and e.
Answer: D
7. {p, q} is a subset of M.{p, q} M
Answer: B
8. QP
P ⊂ QP is a subset of Q.Answer: C
9. Number of subsets of P = 24
= 16
Answer: C
10. P′ is the complement of set P.
ξP
Answer: A
11. E′ = {1, 9}\ Elements of E′ = 1, 9
Answer: D
12. R T= {a, b , c , d, e } { b , c , e , f}= {b, c, e}
Answer: B
13. ξ P
P � Q
Q
Answer: C
14. ξ P QP � Q
Answer: C
15. n(E F G)= number of elements in sets E, F and G= 6
Answer: A
CHAPTER
12 SetsCHAPTER
2
Mathematics SPM Chapter 12
© Penerbitan Pelangi Sdn. Bhd.
16. = E F GE = {A, P, R, I, L}F = {M, A, Y}G = {J, U, L, Y}E F G = { }n(E F G) = 0
Answer: A
17. J = {R, E, A, D}K = {W, R, I, T, E}J K = {R, E, A, D, W, I, T}n(J K) = 7
Answer: C
18. ξ
b c
d
ak f
e
hg
P Q
(P Q)′ = {a, b, e, f, g, h, k}
Answer: D
Paper 2
1. (a) 1
2
3
P
(b) abcd
Q
2. (a) A = {2, 3, 5, 7} \ Elements of A = 2, 3, 5, 7.
(b) B = {a, b, d, k, p} \ Elements of B = a, b, d, k, p
3. PQ
c e
ab
4. A B
5. Number of possible subsets= 23
= 8
{ }, {h}, {i}, {j}, {h, i}, {h, j}, {i, j}, {h, i, j}
6. A
ξ
Complementof set A, i.e. A'
7. Q is a subset of P.P is a subset of ξ.
PQ
ξ
a
b
c
e
d
8. (a) A′ = {6, 12, 18, 21} Elements of A′ = 6, 12, 18, 21
(b) Number of elements in set A′, n(A′) = 4
9. (a) P = {a, b} \ Elements of P = a, b
(b) Q = {a, b, c} \ Elements of Q = a, b, c
(c) Q′ = {d, e} \ Elements of Q′ = d, e
10. (a) A B = {1, 3, 4, 5, 6, 7} \ Elements of A B = 1, 3, 4, 5, 6, 7
(b) A B = {7} \ Element of A B = 7
11. (a) P Q
(b) P Q
3
Mathematics SPM Chapter 12
© Penerbitan Pelangi Sdn. Bhd.
12. (a)
E F
E F
=
E F
(b) EF
EF
=
EF
13. (a) P
Q R
(b) P
Q R
Paper 1
1. n(P < R) = 5 + 6 + 4 + x + 2 + 2x = 17 + 3x
n(Q9) = 5 + 6 + 2x + (2x + 1) = 12 + 4x
n(P < R) = n(Q9) 17 + 3x = 12 + 4x 17 – 12 = 4x – 3x 5 = x
n( ) = 17 + 3x + (2x + 1) = 18 + 5x = 18 + 5(5) = 43Answer: B
2. ξL M
N
ξL M
N
L < M N′
=
ξL M
N
L < M > N′
Answer: D
3. ξ
L
125 – 40 40 92 – 40
28
M
Total number of Form Five students= (125 − 40) + 40 + (92 − 40) + 28= 205Answer: D
4. QP
66 30 x
x + 30 + 66 = 164 x + 96 = 164 x = 164 – 96 = 68 n(Q) = 68 + 30 = 98Answer: C
5.
•d•f
•c•e•g•b•a
E G
F
E′ = {c, d, e, f}Answer: D
4
Mathematics SPM Chapter 12
© Penerbitan Pelangi Sdn. Bhd.
6. ξ EF
ξ EF
E F′
=
ξ E
F
E F′
Answer: B
7. ξ = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} J = {4, 6, 8, 10, 12} K = {3, 4, 6, 12} L = {3, 5, 7, 11, 13} J K = {4, 6, 12}(J K)′ = {3, 5, 7, 8, 9, 10, 11, 13}
(J K)′ L = {3, 5, 7, 8, 9, 10, 11, 13}\ Elements of (J K)′ L = 3, 5, 7, 8, 9, 10, 11, 13
Answer: D
8. {a}, {b}, {a, b}, { }
Answer: D
9. ξP Q
ab
cd
fg
e
Q′ = {a, b, f, g}
Answer: C
10. E , F and F ∩ G ≠ f
E G
F
Answer: A
11. F
E
Answer: A
12. EF G
EF G
E F′ E
F G
G′
=
EF G
E ∩ F′ ∩ G′
Answer: A
13. n(R) = n(P Q)′ 1 + 2 + (x − 5) + (x − 2) = x − 2 3 + 2x − 7 = x − 2 2x − x = −2 − 3 + 7 x = 2
Answer: B
14. S ∩ T= {a, d, e, h} ∩ {a, d, f, g, h}= {a, d, h}
R (S ∩ T) = {a, c, d, f} {a, d, h}= {a, c, d, f, h}
Answer: C
15. (3x + 1) + (x + 5) = 38 4x + 6 = 38 4x = 38 – 6 4x = 32 x = 8Number of players who play two games= 2x= 2(8)= 16
Answer: C
16. ξ = {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} P = {8, 12, 16} Q = {6, 8, 10, 12, 14, 16}All the elements of set P are also elements of set Q.Hence, P , Q
Answer: B
5
Mathematics SPM Chapter 12
© Penerbitan Pelangi Sdn. Bhd.
17. ξ = {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} P = {16, 18, 21, 23, 25, 27} P′ = {17, 19, 20, 22, 24, 26}
Answer: D
18. ξF G
ξF G
F′ G′
=
ξF G
(F G)′
Answer: B
19.
8
1630 26
z
xy
Badminton
BowlingSoccer
x = 69 − (8 + 30 + 16) = 15y = 71 − (x + 16 + 26) = 71 − (15 + 16 + 26) = 14z = 120 − (8 + 30 + 16 + 26 + x + y) = 120 − (8 + 30 + 16 + 26 + 15 + 14) = 11
Number of pupils who played soccer or bowling and also badminton= 8 + x + y= 8 + 15 + 14= 37
Answer: C
Paper 2
1. (a) P Q
Q′
(b) A B
D
A B
D
B (A D)
=
A B
D
B (A D)
2. (a) A DB
A B
(b) A D
B A D
B
A (B D′)
=
A D
B
A (B D′)
3. (a) RQP
P R
(b) RQP
RQP
(P Q′) R
=
RQP
(P Q′) R
6
Mathematics SPM Chapter 12
© Penerbitan Pelangi Sdn. Bhd.
4. (a) A CB
A B
(b) A CB
A CB
(A′ B) C
=
A C
B
(A′ B) C
5. (a) PQ
R
P Q′
(b) PQ
R
PQ
R
(P Q) R′
=
P
QR
(P Q) R′
Paper 1
1. n(G′) = n(E F) (8 − x) + 5 + 6 = 5 + x 8 − x + 11 = 5 + x 8 + 11 − 5 = x + x 14 = 2x
x = 14–––2
= 7
Answer: C
2. ξ E
7 – 5 11 – 55
F
n(E′) = 11 − 5 = 6
Answer: B
3. ξM N
a be
cf
gd
h
(M N)′ = {d, g, h}
Answer: C
4. ξ = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12} E = {3, 5, 7, 9, 11} F = {3, 6, 9, 12}E F = {3, 5, 6, 7, 9, 11, 12}
n(E F) = 7
Answer: C
5. n(P Q R) = 32 4 + 6 + 2 + 3 + 5 + 1 + x = 32 21 + x = 32 x = 32 − 21 = 11
Answer: D
6. F
G H
ξ F
G H
ξ
(F G) H
=
F
G H
ξ
(F G) H
Answer: D
7
Mathematics SPM Chapter 12
© Penerbitan Pelangi Sdn. Bhd.
7.
P
Q Rξ
∩
P
Q Rξ
P′ (Q R)
=
P
Q Rξ
P′ (Q R)
Answer: B
8. M N = {B, L, U, E, P, I, N, K} L′ = {U, E, P, I, N}(M N) L′ = {U, E, P, I, N}
Answer: C
9. ξ X ZY
(X Y)
ξ X ZY
Z′
=
ξ X ZY
(X Y) Z′
Answer: A
10. ξP
Q
∩
ξP
Q
Q P′= φ (empty set)
Answer: D
Paper 2
1. (a) GFE
E F
(b) GFE
GFE
(E F)′ G
=
GFE
(E F)′ G
2. (a) P R
Q
P Q
(b) P R
Q
P RQ
(P Q) R
=
P R
Q
(P Q) R
3. (a) A B
C
A B′
8
Mathematics SPM Chapter 12
© Penerbitan Pelangi Sdn. Bhd.
(b) A BC
A BC
(A B) C′
=
A B
C
(A B) C′
4. (a) X Y
Z
X Y′
(b) X Y
Z
X Y
Z
(X Y) Z
=
X Y
Z
(X Y) Z
5. (a) Q
PR
P R′(b)
QP
R QP
R
(Q R) P′
=
Q
PR
(Q R) P′
6. (a) E GF
E′ G′
(b) E GF
E GF
(E F) (F G)
=
E GF
(E F) (F G)
7. (a) P
Q R
(P Q)′
(b) P
Q R
P
Q R (P Q)′ R
=
P
Q R
(P Q)′ R
8. (a) A
BC
B C′
9
Mathematics SPM Chapter 12
© Penerbitan Pelangi Sdn. Bhd.
(b) ABC
ABC
(A B′) C
=
A
BC
(A B′) C
9. (a) Let x be the number of students who study Physics. Then 2x be the number of students who study Chemistry.
4
856
21 – 2 – 5 – 8 = 6 2
Physics
Biology
Chemistry
x 2x
(b) x + 2x + 4 = 43 3x = 39 x = 13 \ 13 students.
10. x + 6y + 10 = 58 x + 6y = 48 …… 1 1
4 (3y + y + 6y + 10) = 10 + y
10y + 10 = 40 + 4y 10y – 4y = 40 – 10 6y = 30 y = 5
From 1, x + 6(5) = 48 x = 48 – 30 = 18
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