Tg4_praktis_4

4BAB Modul
Operasi Set PBD
Operations on Sets

4.1 Persilangan Set/ Intersection of Sets

1 Selesaikan setiap yang berikut. (a) Diberi/Given P = {m, a, n, i, s}, Q = {p, a, h, i,
Solve each of the following. SP 4.1.1 TP 2 t} dan/and R = {a, e, i, o, u}
Cari/Find
CONTOH (i) P ∩ Q
Diberi/Given P = {1, 2, 3, 4, 5}, Q = {3, 5, 7, 9} P ∩ Q = {a, i}
dan/and R = {2, 4, 6, 7}
Cari/Find (ii) Q ∩ R
Q ∩ R = {a, i}
(i) P ∩ Q
P ∩ Q = {3,5} (iii) P ∩ R
P ∩ R = {a, i}
(ii) Q ∩ R
Q ∩ R = {7}

(iii) P ∩ R
P ∩ R = {2, 4}

Tip Pintar

Cari unsur sepunya daripada kedua-dua set.
Find the common elements from both sets.

2 Selesaikan yang berikut. (i) M = {2, 3, 5, 7, 11, 13}
Solve the following. SP 4.1.1 TP 3 N = {1, 3, 5, 7, 9, 11, 13, 15}
M ∩ N = {3, 5, 7, 11, 13}
CONTOH
(ii)
Diberi/Given ξ = {x : 1 ≤ x ≤ 15},
M = {x : x ialah nombor perdana/x is a prime M N 12
2 31 14
number}
N = {x : x ialah nombor ganjil/x is an odd number} 59
7
(i) Cari/Find M ∩ N 11 15
(ii) Lukis Venn diagram untuk mewakili ξ, M 13

dan N. 46 10
Kemudian, lorekkan M ∩ N. 8
Draw the Venn diagram to represent ξ, M
and N. Hence, shade the region M ∩ N.

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(a) Diberi/Given ξ = {x : 8 ≤ x ≤ 15}, (i) M ∩ N
M = {x : x ialah gandaan 2/x is a multiple of 2} M = {8, 10, 12, 14}
N = {x : x ialah gandaan 3/x is a multiple of 3} N = {9, 12, 15}

Cari/Find M ∩ N = {12}
(i) M ∩ N
(ii) Lukis Venn diagram untuk mewakili ξ, M (ii)
dan N. Kemudian,lorekkan M ∩ N.
Draw the Venn diagram to represent ξ, M
and N. Hence, shade the region M ∩ N.

3 Selesaikan yang berikut
Solve the following SP 4.1.1 TP 3

CONTOH (a) Diberi/Given = {10, 11, 12, 13, 14, 15, 16, 17,
18}, A = {10, 11, 15, 18}, B = {13, 14, 15, 16,
Diberi/Given ξ = {1, 2, 4, 6, 8, 9, 10}, 18}, C = {11, 12, 14, 15, 17}.
A = {1, 3, 4, 10}, B = {2, 4, 6, 8, 10}, (i) Cari/Find
C = {1, 2, 4, 8, 9}. (1) A ∩ B
= {15, 18}
(a) Cari/Find (2) B ∩ C
(i) A ∩ B = {14, 15}
= {4,10} (3) A ∩ B ∩ C
(ii) B ∩ C = {15}
= {2,4,8} (ii) Lukis dan lorekkan Venn diagram yang
(iii) A ∩ B ∩ C mewakili A ∩ B ∩ C. /Draw and shade the
= {4} Venn diagram that represent A ∩ B ∩ C.

(b) Lukis dan lorekkan Venn diagram yang (iii) Nyatakan hubungan antara
mewakili A ∩ B ∩ C./Draw and shade (1) B ∩ C dan A
the Venn diagram that represent B∩C⊄A
A ∩ B ∩ C. (2) A ∩ B ∩ C dan A
A∩B∩C⊂A
A B
3 10 6

4
12

8

9
C

(c) Nyatakan hubungan antara
(i) A ∩ B dan A
A∩B⊂A

(ii) A ∩ B ∩ C dan B
A∩B∩C⊂B

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(b) Diberi/Given ξ = {x : 11 ≤ x ≤ 20}, A = {11, 13, (i) (1) {11, 13, 17, 19}
15, 17, 19}, B = {11, 13, 17, 19}, C = {13}. (2) {13}
(3) {13}
(i) Cari/Find
(1) A ∩ B (ii)
(2) B ∩ C
(3) A ∩ B ∩ C

(ii) Lukis dan lorekkan Venn diagram yang
mewakili A ∩ B ∩ C./Draw and shade the
Venn diagram that represent A ∩ B ∩ C.

4 Lorekkan rantau yang mewakili set-set berikut.
Shade the region that represent the following sets. SP 4.1.2 TP 2

CONTOH (a) A ∩ B’ B (b) A ∩ B
A∩B
A A
A B B

(c) A ∩ B ∩ C (d) (A ∩ B ∩ C)’ (e) A ∩ B ∩ C’ B

A BA BA
C

CC

Ingat Lagi?

Set pelengkap B diwakili oleh B’ mempunyai unsur-unsur yang tidak ada dalam B.
The complement of B is represented by B’ having elements that is not in B.

5 Lorekkan rantau yang mewakili set-set berikut.
Shade the region that represent the following sets. SP 4.1.2 TP 2

CONTOH (a) A’ ∩ B’ (b) A ∩ B’
A ∩ B ∩ C’
BA
A B
AB

C

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(c) (A ∩ B) ∩ C’ (d) A ∩ B ∩ C’ (e) (A ∩ B)’ ∩ C B

A BA A
B

C

C
C

6 Gambar rajah Venn menunjukkan unsur-unsur dalam beberapa set. Cari persilangan set berikut.
The Venn diagram shows the elements in a few sets. Find the following intersection sets. SP 4.1.2 TP 3

CONTOH d Q h (a) 3 Q
e i 5
P f ξ 2
a g 1 6
P 47
b
c

(i) P ∩ Q (i) P ∩ Q
= {d,e} = {4}

(ii) P ∩ Q’ (ii) P ∩ Q’
= {a,b,c} = {1, 2}

(iii) P’ ∩ Q (iii) P’ ∩ Q
= {f,g,h,i} = {5, 6, 7}

(iv) (P ∩ Q)’
= {1, 2, 3, 5, 6, 7}

(b) (c)

␰ 90 ␰ Q
P
PR Q
40 bf
60
10 20 50 70 g
c

eh

80 di
30

jk a
R

(i) P ∩ Q ∩ R (i) P ∩ Q ∩ R
= {20} = {e}

(ii) P ∩ Q’ ∩ R (ii) P ∩ (Q’ ∩ R)
= {30} = {d}

(iii) P’ ∩ Q ∩ R’ (iii) P’ ∩ (Q ∩ R)’
= {60, 70, 80} = {a, f, g, h, j, k}

(iv) (P ∩ Q ∩ R)’ (iv) (P ∩ Q ∩ R)’
= {10, 30, 40, 50, 60, 70, 80, 90} = {a, b, c, d, f, g, h, i, j, k}

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7 Selesaikan masalah yang berikut.
Solve the following problem. SP 4.1.3 TP 4

CONTOH

Dalam sekumpulan pekerja seramai 40 orang, 25 orang suka minum teh, 30 orang suka minum
kopi dan 4 orang tidak suka kedua-dua jenis minuman. Berapa orang yang suka

(i) kedua-dua jenis minuman?
(ii) teh sahaja?
In a group of 30 workers, 25 like tea, 30 like coffee and 4 like neither. How many like
(i) both drinks?
(ii) only tea?

␰

TK

25 – x x 30 – x

4

(i) Biarkan/Let n(T ∩ K) = x

n(T ∩ K’) = 25 − x
n(K ∩ T’) = 30 – x

n(ξ) = 40
25 – x + x + 30 – x + 4 = 40

x = 19

(ii) n(T ∩ K’) = 25 – x
= 25 – 19
=6

Ingat Lagi?

n(T ∩ K) mewakili bilangan unsur dalam persilangan set T dan K.
n(T ∩ K) represent the number of elements in the intersection of sets T and K.

(a) Jadual menunjukkan bilangan pelajar yang lulus bagi subjek Bahasa Inggeris, Sains dan Matematik
dalam sebuah kelas. Jumlah pelajar di dalam kelas itu ialah 42 orang. Setiap pelajar lulus
sekurang-kurangnya satu subjek.

The table shows the number of students who pass English, Science and Mathematics subjects in a class.

There are 42 students in the class. Each student pass at least one subject.

Subjek/Subjects Bilangan pelajar
The number of
Bahasa Inggeris sahaja/English only
Sains sahaja/Science only students
Matematik sahaja/Mathematics only
Bahasa Inggeris dan Sains sahaja/English and Science only x
Sains dan Matematik sahaja/Science and Mathematics only
Bahasa Inggeris dan Matematik sahaja/English and Mathematics only 8
Bahasa Inggeris, Sains dan Matematik/English, Science and Mathematics
13

0

3

5

9

(i) Wakilkan nilai-nilai di dalam jadual dengan Venn diagram dan cari nilai x.
Represent the values in the table using Venn diagram and find the value of x.

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(ii) Cari bilangan pelajar yang lulus Sains dan Matematik.
Find the number of students who pass Science and Mathematics.

(i)

x + 8 + 13 + 5 + 3 + 9 = 42
x=4

(ii) n(S ∩ M) = 9 +3 = 12 orang

4.2 Kesatuan Set/ Union of Sets (a) Diberi/Given P = {11, 12, 13, 14, 15},
Q = {11, 13, 15} dan/and R = {12, 14, 16}
1 Selesaikan yang berikut.
Solve the following. SP 4.2.1 TP 2 Cari/Find

CONTOH (i) P ∪ Q
Diberi/Given P = {a, e, i, o, u}, Q = {g, e, n, P ∪ Q = {11, 12, 13, 14, 15}
a, p} dan/and R = {g, a, n, j, i, l}
Cari/Find (ii) Q ∪ R
(i) P ∪ Q Q ∪ R = {11, 12, 13, 14, 15, 16}
P ∪ Q = {a, e, g, i, n, o, p, u}
(ii) Q ∪ R (iii) P ∪ R
Q ∪ R = {a, e, g, i, j, l, n, p} P ∪ R = {11, 12, 13, 14, 15, 16}
(iii) P ∪ R
P ∪ R = {a, e, g, i, j, l, n, o, u}
Tip Pintar

Unsur yang sama tidak perlu diulang.
The same elements should not be repeated.

2 Selesaikan yang berikut. (i) M = {2, 4, 6, 8, 10, 12}
Solve the following. SP 4.2.1 TP 3
N = {3, 6, 9, 12}
CONTOH
M ∪ N = {2, 3, 4, 6, 8, 9, 10, 12}
Diberi/Given ξ = {x : 2 ≤ x ≤ 12}, (ii)
M = {x : x ialah nombor genap/x is an even
number} ξ 4 N
N = {x : x ialah gandaan 3/x is a multiple of 3} M 8 3
69
(i) Cari/Find M ∪ N
(ii) Lukis Venn diagram untuk mewakili ξ, M 2 12

dan N. Kemudian,lorekkan M ∪ N. 10
Draw the Venn diagram to represent ξ, M
and N. Hence, shade the region M ∪ N.

57 11

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(a) Diberi/Given ξ = {x : 1 ≤ x ≤ 12}, (i) M ∪ N
M = {x : x ialah faktor bagi 10/x is a factor of M = {1, 2, 5, 10}
10} N = {1, 2, 3, 4, 6, 12}
N = {x : x ialah faktor bagi 12/x is a factor of M ∪ N = {1, 2, 3, 4, 5, 6, 10, 12}
12}
(ii)
Cari/Find

(i) M ∪ N
(ii) Lukis Venn diagram untuk mewakili ξ, M

dan N. Kemudian,lorekkan M ∪ N.
Draw the Venn diagram to represent ξ, M
and N. Hence, shade the region M ∪ N.

3 Selesaikan yang berikut. (b) Diberi/Given ξ = {34, 35, 36, 37, 38, 39, 40,
Solve the following. SP 4.2.1 TP 3 41, 42},
A = {34, 35, 36, 37, 41}, B = {36, 37, 38, 39, 40},
(a) Diberi/Given ξ = {x : 1 ≤ x ≤ 10}, C = {37, 38, 41, 42}.
A = {1, 3, 4, 7, 10}, B = {2, 4, 5, 6, 8, 10},
C = {1, 2, 4, 8, 9}. (i) Cari/Find
(1) A ∪ B
(i) Cari/Find = {34, 35, 36, 37, 38, 39, 40, 41}
(1) A ∪ B (2) B ∪ C
= {1, 2, 3, 4, 5, 6, 7, 8, 10} = {36, 37, 38, 39, 40, 41, 42}
(2) B ∪ C (3) A ∪ B ∪ C
= {1, 2, 4, 5, 6, 8, 9, 10} ={34, 35, 36, 37, 38, 39, 40, 41, 42}
(3) A ∪ B ∪ C
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} (ii) Lukis dan lorekkan Venn diagram yang
mewakili A ∪ B ∪ C./Draw and shade the
(ii) Lukis dan lorekkan Venn diagram yang Venn diagram that represent A ∪ B ∪ C.
mewakili A ∪ B ∪ C./Draw and shade the
Venn diagram that represent A ∪ B ∪ C.

(c) Diberi/Given ξ = {x : 11 ≤ x ≤ 20}, (i) (1) {11, 12, 13, 14, 15, 16, 17, 18, 20}
A = {11, 12, 13, 14, 15, 20}, B = {13, 14, 15, (2) {13, 14, 15, 16, 17, 18}
16, 17, 18}, (3) {11, 12, 13, 14, 15, 16, 17, 18, 20}
C = {16}.
(ii)
(i) Cari/Find
(1) A ∪ B ␰
(2) B ∪ C
(3) A ∪ B ∪ C

(ii) Lukis dan lorekkan Venn diagram yang
mewakili A ∪ B ∪ C./Draw and shade the
Venn diagram that represent A ∪ B ∪ C.

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4 Lorekkan rantau yang mewakili set-set berikut.
Shade the region that represent the following sets. SP 4.2.2 TP 2

CONTOH (a) A ∪ B’ B (b) A ∪ B
A∪B
A A
A B

B

(c) A ∪ B ∪ C (d) A’ ∪ B ∪ C (e) A ∪ B ∪ C’ B

A BA BA
C

CC

5 Lorekkan rantau yang mewakili set-set berikut.
Shade the region that represent the following sets. SP 4.2.2 TP 2

CONTOH (a) A’ ∪ B’ (b) (A ∪ B)’
A ∪ B ∪ C’
ξ ξ
ξA A
B AB

B
C

(c) (A ∪ B) ∪ C’ (d) A ∪ B’ ∪ C (e) (A ∪ B)’ ∪ C

ξA Bξ A ξ B

BA
C

C
C

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6 Senaraikan unsur-unsur kesatuan set yang berikut berpandukan gambar rajah Venn yang diberi.
List all elements of the following union of sets based on the given Venn diagram. SP 4.2.2 TP 3

CONTOH Q (a) Q
55
Pa ξ
b
c df h 33
P
e gi
11 66
22
44 77

(i) P ∪ Q (i) P ∪ Q
= {a, b, c, d, e, f, g, h, i} = {11, 22, 44, 55, 66, 77}

(ii) P ∪ Q’ (ii) P ∪ Q’
= {a, b, c, d, e, f, g} = {11, 22, 33, 44}

(iii) P’ ∪ Q (iii) P’ ∪ Q
= {d, e, f, g, h, i} = {33, 44, 55, 66, 77}

(iv) (P ∪ Q)’
= {33}

(b) Q (c) 41

ξ ξ

P P Q
31 38
40 60 32 42
10 50 70 R
80 20 39
30
90 36 40

33 37

(i) P ∪ Q ∪ R 34
= {10, 20, 30, 40, 50, 60, 70, 80} R 35

(ii) P ∪ Q’ ∪ R (i) P ∪ Q ∪ R
= {10, 20, 30, 40, 50, 80, 90} = {31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
42}
(iii) P’ ∪ Q ∪ R’
= {10, 20, 30, 40, 50, 60, 70, 80, 90} (ii) P ∪ (Q’ ∪ R)
= {31, 32, 33, 34, 35, 36, 37, 41, 42}
(iv) (P ∪ Q ∪ R)’
= {90} (iii) P’ ∪ (Q ∪ R)’
= {31, 32, 34, 35, 37, 38, 39, 40, 41}

(iv) (P ∪ Q ∪ R)’
= {41}

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7 Selesaikan masalah yang berikut.
Solve the following problem. SP 4.2.3 TP 4

(a) Dalam sekumpulan pekerja seramai 30 orang, 12 orang suka makan nasi lemak dan roti canai.

Bilangan pekerja yang suka nasi lemak adalah dua kali ganda daripada pekerja yang suka roti
canai. Jika 3 orang tidak suka kedua-duanya, berapakah bilangan pekerja yang hanya suka makan
roti canai?
In a group of 30 workers, 12 like to eat nasi lemak and roti canai. The number of workers who like nasi
lemak is twice the workers who like roti canai. If 3 workers like neither, how many workers like only roti
canai?

Katakan/Let x = bilangan pekerja yang suka roti canai/number of workers who like roti canai
Bilangan pekerja yang suka nasi lemak/The number of worker who like nasi lemak = 2x
Katakan/Let ξ = {bilangan pekerja/the number of workers}
N = {pekerja yang suka nasi lemak/the worker who like nasi lemak}
R = {pekerja yang suka roti canai/the worker who like roti canai}

ξ

n(ξ) = 30
n(N ∪ R) + 3 = 30

2x – 12 + 12 + x – 12 + 3 = 30
39
x = 3

= 13

Bilangan pekerja yang suka makan roti canai sahaja/The number of workers who like only roti canai
x – 12 = 13 – 12

=1

(b) Gambar rajah Venn menunjukkan bilangan kanak-kanak di sebuah tadika. Set B = {kanak-kanak
yang suka biru}, set H = {kanak-kanak yang suka hijau} dan set K = {kanak-kanak yang suka
kuning}. Di beri n(H ∪ K) = 20, cari bilangan kanak-kanak yang

(i) suka warna biru atau hijau.
(ii) suka warna biru atau tidak suka warna kuning.

The Venn diagram shows the number of children in a kindergarten. Set B = {children who like blue}, set

H = {children who like green} and set K = {children who like yellow}. Given n(H ∪ K) = 20, find the
number of children who

(i) like blue or green. Diberi/Given n(H ∪ K) = 20
(ii) like blue or do not like yellow. 2x + 10 = 20

B H K x=5
10 x
4 (i) n(B ∪ H) = 10 + x + 4 + 6
x6 = x + 20
= 5 + 20
= 25

(ii) n(B ∪ K’) = 10 + x + 4
= 10 + 5 + 4
= 19

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4.3 Gabungan Operasi Set/ Combined Operation of Sets

1 Diberi/Given ξ = {x: x ialah gandaan 5 yang kurang dari 30/x is a multiple of 5 less than 30},
A = {5, 10, 15}, B = {15, 20, 25} dan/and C = {15}.

Cari set yang berikut.
Find the set of the following. SP: 4.3.1 & 4.3.2 TP 2

CONTOH (a) A ∩ B ∪ C (b) A ∪ (B ∩ C)
A ∪ B ∩ C’ = {15} = {5, 10, 15}
= {5, 10, 20, 25}

(c) A ∪ (B ∩ C)’ (d) (A ∪ B) ∩ (B ∩ C) (e) A ∩ (B ∪ C’)
= {5, 10, 20, 25} = {15} = {5, 10, 15}

2 Selesaikan yang berikut.
Solve the following. SP 4.2.1 TP 3

(a) Diberi/Given (b) Diberi/Given
ξ = {11, 12, 13, 14, 15, 16, 17, 18, 19, 20} ξ = {x : 1 ≤ x ≤ 12 }
A = {11, 13, 17, 19} A = {1, 2, 3, 4, 6}
B = {11, 13, 15, 17, 19} B = {2, 5, 9}
C = {12, 14, 16, 18, 20} C = {2, 4, 8, 10, 12}
Lukis Venn diagram untuk mewakili ξ, A, B
Lukis Venn diagram untuk mewakili ξ, A, B dan C./Draw Venn diagram to represent A, B and
dan C./Draw Venn diagram to represent A, B and C.
C
(i) Lorek/Shade (A ∩ C) ∪ C
(i) Lorek/Shade A ∩ B ∪ C

(ii) Lorek/Shade (A ∩ B)’ ∪ C

(ii) Lorek/Shade A ∩ (B ∪ C’)

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3 Gambar rajah Venn menunjukkan unsur dalam set P, Q dan R. Selesaikan masalah yang berikut.
The Venn diagram shows the element in set P, Q and R. Solve the following problems. SP: 4.3.1 & 4.3.2 TP 3

ξ P Q 1 (a) P ∩ Q ∪ R = {2, 4, 9}
3 n(P ∩ Q ∪ R) = 3
R 7 4 5
9 8 2 6 (b) P ∪ Q ∩ R = { }
n(P ∪ Q ∩ R) = 0
10
(c) P ∪ (Q ∩ R) = {2, 4, 7, 8, 10}
n(P ∪ (Q ∩ R)) = 5

(d) P’ ∩ Q ∪ R = {9}
n(P’ ∩ (Q ∪ R)) = 1

(e) P ∩ Q’ ∪ R = {7, 8, 9, 10}
n(P ∩ Q’ ∪ R) = 4

(f) P ∩ (Q ∪ R)’ = {7, 8, 10}
n(P ∩ (Q ∪ R)’) = 3

(g) (P ∪ Q ∪ R)’ = {1, 3, 5, 6}
n(P ∪ Q ∪ R)’ = 4

(h) (P ∩ Q) ∪ R = {2, 4, 9}
n((P ∩ Q) ∪ R) = 3

(i) (P ∪ Q) ∩ R’ = {2, 4, 7, 8, 10}
n((P ∪ Q) ∩ R’) = 5

4 Tulis suatu ungkapan bagi set yang diwakili oleh kawasan berlorek dengan menggunakan simbol ∪ dan
∩.
Write down an expression for the set indicated by the shaded region using the symbol ∪ and ∩. SP: 4.3.1 & 4.3.2 TP 5

CONTOH (a) (b)

A BA B A
C
B

C (A ∩ B) ∪ (A ∪ B)’ C

A ∪ (B’ ∩ C) A’ ∪ B ∪ C

(c) (d) B (e) B

A A A
B CC C

(A ∪ B) ∩ C’ A ∩ B ∪ C’

(A ∩ B) ∪ C

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5 Selesaikan masalah yang berikut.
Solve the following problem. SP 4.3.3 TP 5

(a) Gambar rajah Venn menunjukkan bilangan (b) Gambar rajah Venn menunjukkan bilangan
murid yang menyertai persatuan pidato, P, murid dalam set E, O dan A. Diberi set E =
persatuan robotik, R dan persatuan sains, murid yang sukakan epal, set O = murid yang
S. Setiap murid mesti menyertai sekurang- sukakan oren dan set A = murid yang sukakan
kurangnya satu penyertaan. anggur. Setiap murid sukakan sekurang-
The Venn diagram shows the number of students kurangnya satu jenis buah.
joining the public speaking club, P, robotics club, The Venn diagram shows the number of students
R and science club, S. Each student must join at in set E, O and A. Given set E = students who like
least one club. apple, set O = students who like orange and set
A = students who like grapes.
Diberi/Given: ξ = P ∪ R ∪ S, n(S) = 36, cari/
find: Diberi/Given:
ξ = E ∪ O ∪ A dan/and n(ξ) = 45

R P E O
8 15 3 x+2
x 15
x
2 22
3

x+9 3
2
S x

(i) bilangan murid yang menyertai satu A
persatuan sahaja,
the number of students who join one club (i) bilangan murid yang sukakan ketiga-tiga
jenis buah,
only, the number of students who likes all the three

n(S) = 36 types of fruits,
x + 9 + x + 2 + 3 = 36
2x + 14 = 36 n(E ∩ O ∩ A) = x
x = 11
15 + 3 + x + 2 + 2 + x + 2 + 3 x = 45
2

Bilangan murid yang menyertai satu 7 x + 24 = 45
persatuaaan sahaja/The number of students 2

who joins one club only x=6

=8+x+9+5 (iii) bilangan murid yang sukakan epal atau
= 8 + 11 + 9 + 5 oren tetapi tidak suka anggur,
= 33 the number of students who like apple or

(ii) bilangan murid yang menyertai persatuan orange but do not like grapes,
Robotik,
the number of students who join Robotics n(E ∪ O ∩ A’) = 15 + 3 + x + 2
= 26
club,
(iii) bilangan murid yang sukakan oren dan
=x+8+2+1 anggur.
= 11 + 8 + 2 + 1 the number of students who like oranges and
= 22 grapes.

(iii) bilangan murid yang menyertai dua =2+x
persatuan sahaja. =8
the number of students who join one club

only.

=1+x+3
= 1 + 11 + 3
= 15

© Oxford Fajar Sdn. Bhd. (008974-T) 2019 57

Praktis Pentaksiran 4

Kertas 1

1 Gambar rajah menunjukkan set ξ, A, B dan C. 4 Gambar rajah Venn menunjukkan unsur
dalam set A, B dan C.
Klon Diagram shows the sets ξ, A, B and C . Venn diagram shows the elements in set A, B and
SPM
ξ‘15 C.

AB B
C

A1 11 C
23 8

4 7 10 12
5 13

Kawasan berlorek mewakili 9 14
The shaded region represents
A (A ∩ B) ∩ C’ Diberi bahawa set semesta, ξ = A ∪ B ∪ C.
B (A ∪ B) ∩ C’ Senaraikan unsur bagi set A ∪ B ∩ C.
C (A ∪ B)’ ∩ C’
D A ∪ B’ Given universal set, ξ = A ∪ B ∪ C. List all the
elements of set A ∪ B ∩ C.
2 Gambar rajah Venn menunjukkan bilangan A {10}
unsur dalam set A, B dan C. B {7, 3}
Venn diagram shows the number of elements in C {1, 2, 4, 5, 10}
D {11, 12, 13, 14}
set A, B and C.

AB C 5 Gambar rajah Venn menunjukkan bilangan
3 11 unsur dalam set R, S dan T.
Venn diagram shows the number of elements in
4
set R, S and T.

Diberi bahawa set semesta, ξ = A ∪ B ∪ C. R
Cari nilai n(A ∩ B)’. S

Given universal set, ξ = A ∪ B ∪ C. Find the T

value of n(A ∩ B)’.
A3 C 18
m+1 m 17
B 15 D 20

3. Diberi ξ = {x: x adalah integer, 5 ≤ x ≤ 25} 2m
Given ξ = {x: x is an integer , 5 ≤ x ≤ 25}
Set A = {x: x adalah faktor bagi 20} Diberi bahawa set semesta, ξ = R ∪ S ∪ T.
Set A = {x: x is a factor of 20} dan n(R) = n(T ∩ R’), tentukan nilai x.

Set B ={x: x adalah faktor bagi 25} Given universal set, ξ = R ∪ S ∪ T and
Set B = {x: x is a factor of 25} n(R) = n(T ∩ R’), determine the value of m.
A3
Pernyataan yang manakan benar? B4
C5
Which of the following statements is true? D8

A A⊂B C (A ∩ B) ⊂ B
B B⊂A D (A ∩ B) = B

© Oxford Fajar Sdn. Bhd. (008974-T) 2019 58

6 Gambar rajah Venn menunjukkan bilangan Antara yang berikut, yang manakah mewakili

Klon unsur dalam set P, set Q dan set R dengan set kawasan berlorek?
SPM
‘14 semesta, ξ = P ∪ Q ∪ R. Which of the following represents the shaded

The Venn diagram shows the number of elements region?

in sets P, Q and R where the universal set ξ = P ∪ A (A ∩ B)’ ∩ C C A’ ∩ (B ∩ C)
B (A ∩ B)’ ∪ C D A’ ∩ (B ∪ C)
Q ∪ R.

P Q R 10 Gambar rajah Venn menunjukkan bilangan
31 15
6 + 3y Klon ahli bagi tiga kelab sukan.
y 1 SPM
‘17 The following Venn diagram shows the number of

members of three sports club.

Di beri n(P) = n(Q ∪ R). Cari nilai y. Kelab Bola Jaring Kelab Bola Sepak
Netball Club Football Club
Given n(P) = n(Q ∪ R). Find the value of y.
A3 C 11 32 5x

B9 D 18 Kelab Hoki x 16
Hockey Club 60

7 Diberi bahawa n(ξ) = 20, P dan Q ialah dua
set dengan keadaan n(P) = 12, n(Q) = 9 dan

n(P ∩ Q) = 4. Cari n(P ∩ Q’).

Given that n(S) = 20, P and Q are two sets such Diberi bahawa bilangan ahli kelab Hoki

that n(P) = 12, n(Q) = 9 and n(P ∩ Q) = 4. Find adalah 1 bilangan ahli Kelab Bola Jaring.
5
n(P ∩ Q'). C6
A1
Hitung bilangan ahli yang menyertai Kelab
B4 D8
Bola Sepak sahaja.

8 Dalam gambar rajah Venn, Given that the number of members of the Hockey
In a Venn diagram,
ξ=P∪Q∪R Club is 1 of the number of members of the
5

PQ Netball Club. Calculate the number of members

who join Football Club only.

R A 30 C 136

AB CD B 35 D 148

11 Gambar rajah Venn menunjukkan bilangan

Klon unsur dalam set A, set B dan set C dengan set
SPM
‘18 semesta, ξ = A ∪ B ∪ C.
Kawasan yang berlabel manakah, A, B, C dan
D mewakili set P ∩ Q ∩ R’? The Venn diagram shows the number of elements
Which region, A, B, C and D represent set P ∩ Q
in sets A, B and C where the universal set ξ = A
∩ R’?
∪ B ∪ C.

B

9 Gambar rajah Venn menunjukkan set A, set B 14
AC
Klon dan set C dengan set semesta, ξ = A ∪ B ∪ C.
SPM 9 10
‘15 The Venn diagram shows sets A, B and C where

the universal set ξ = A ∪ B ∪ C.

AB 3x 28
25 +2x

Di beri n(A) = n(C). Cari nilai x.

Given n(A) = n(C). Find the value of x.

A1 C3

C B2 D4

© Oxford Fajar Sdn. Bhd. (008974-T) 2019 59

Kertas 2

1 Gambar rajah Venn di ruang jawapan menunjukkan set-set P, Q dan R dengan keadaan set universal,

Klon ξ = P ∪ Q ∪ R. Pada rajah di ruang jawapan, lorek
SPM
‘17 The Venn diagram in the answer space shows sets P, Q and R such that the universal set, ξ = P ∪ Q ∪ R. On

the diagram provided in the answer space, shade

(a) set A’
the set A’

(b) set (A ∩ B) ∪ C
the set (A ∩ B) ∪ C

(a)

AB
C

(b)

AB
C

2 Gambar rajah Venn menunjukkan sejumlah 36 kanak-kanak yang meminati beberapa jenis aiskrim.
The Venn diagram shows a total of 36 children that likes a few types of ice cream.

V C
5 X3

2
73

X
S

Set semesta, ξ = V ∪ C ∪ S, di mana
Set V = {kanak-kanak yang meminati ais krim vanilla}
Set C = {kanak-kanak yang meminati ais krim coklat}
Set S = {kanak-kanak yang meminati aiskrim strawberi}

Universal set, ξ = V ∪ C ∪ S, where
Set V = {children who like vanilla ice cream}
Set C = {children who like chocolate ice cream}
Set S = {children who like strawberry ice cream}

© Oxford Fajar Sdn. Bhd. (008974-T) 2019 60

Nyatakan bilangan kanak-kanak yang
(a) meminati ketiga-tiga jenis aiskrim
(b) meminati satu jenis aiskrim sahaja
(c) meminati aiskrim vanilla atau coklat

State the number of children that
(a) like all three types of ice cream
(b) like one type of ice cream only
(c) like vanilla or chocolate ice cream

(a) 2 orang/children
(b) n(V ∪ C ∪ S) = 36

5 + 3 + x + x + 7 + 3 + 2 = 36
2x + 20 = 36
x=8
Bilangan kanak-kanak yang meminati satu jenis aiskrim sahaja/The number of children who like one
type of ice cream only
=5+3+x
=8+8
= 16
(c) n(V ∪ C) = 5 + 3 + 7 + 3 + 2 + x
= 20 + 8
= 28

3 Gambar rajah Venn di ruang jawapan menunjukkan set-set P, Q dan R dengan keadaan set universal,
ξ = P ∪ Q ∪ R. Pada rajah di ruang jawapan, lorek
The Venn diagram in the answer space shows sets P, Q and R such that the universal set, ξ = P ∪ Q ∪ R. On
the diagram provided in the answer space, shade

(a) set Q’ ∩ P
the set Q’ ∩ P

(b) set (P ∩ Q) ∪ R
the set (P ∩ Q) ∪ R

(a)

PQ
R

(b)

R

PQ

© Oxford Fajar Sdn. Bhd. (008974-T) 2019 61

Genius Mengaplikasi

Jadual menunjukkan bilangan pelajar yang lulus subjek Matematik, Sains dan Bahasa Inggeris untuk satu
peperiksaan.

The table shows the number of students who pass Mathematics, Science and English in an examination.

Subjek Bilangan pelajar
Subject Number of student

Matematik 60
Mathematics
58
Sains
Science 52

Bahasa Inggeris 43
English
38
Matematik dan Sains
Mathematics and Science 40

Matematik dan Bahasa Inggeris 31
Mathematics and English

Sains dan Bahasa Inggeris
Science and English

Matematik, Sains dan Bahasa Inggeris
Mathematics, Science and English

(a) Lukis gambar rajah Venn yang mewakili maklumat di atas.
Sketch a Venn diagram to represent the information above.

(b) Cari bilangan pelajar yang tidak lulus Bahasa Inggeris atau Matematik.
Find the number of students who do not pass English or Mathematics.

(c) Cari bilangan pelajar yang lulus Sains dan Matematik sahaja.
Find the number of students who pass Science and Mathematics only.

(a)

(b) n(M ∩ E)’ = 80 – 38 = 42
(c) 12

© Oxford Fajar Sdn. Bhd. (008974-T) 2019 62