Modul Maths Tg 4 Bab 4

Modul • Matematik Tingkatan 4

4UNIT OPERASI SET
OPERATIONS ON SETS

4.1 Persilangan Set / Intersection of Sets

NOTA / NOTES

1 Persilangan set A dan set B ialah suatu set yang mengandungi semua unsur sepunya set A dan
set B.

The intersection of set A and set B is a set containing all common elements in both set A and set B.

2 Jika A ⊂ B, maka A ∩ B = A.

If A ⊂ B, then A ∩ B = A.

3 Persilangan set A dan set B ditulis sebagai A ∩ B.

The intersection of set A and set B is written as A ∩ B.

Contoh / Example:

ξ ξ B
A BA

A∩B A ∩ B =φ UNIT 4

SP4.1.1 Menentukan dan menghuraikan persilangan set menggunakan pelbagai perwakilan.

Determine and describe the intersection of sets using various representations.

NOTA / NOTES

Dalam menentukan dan menghuraikan persilangan set, pelbagai perwakilan boleh digunakan,
seperti

In determining and describing the intersection of the set, various representions can be used, such as

(i) perwakilan secara perihalan,

representation by description,

(ii) perwakilan secara simbolik,

representation by symbolic,

(iii) perwakilan secara grafik.

representation by graphic.

1 Wakilkan persilangan set di bawah dengan menggunakan perwakilan secara perihalan.

Represent the intersection set below by using representation by description.

Kumar telah mengelaskan bulan-bulan dalam setahun seperti di bawah:

Kumar has classified the months of the year as below:

Set ξ  ialah bulan dalam setahun. / Set ξ is the month of the year.
Set A  ialah bulan-bulan yang terdiri daripada 7 huruf. / Set A is the month consisting of 7 letters.
Set B  ialah bulan yang mempunyai 31 hari. / Set B is the month that has 31 days.
Set C  ialah bulan yang mempunyai 28 atau 29 hari sahaja. / Set C is the month that has 28 or 29 days only.
Set D  ialah bulan yang bermula dengan huruf J. / Set D is the month that begins with letter J.

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Modul • Matematik Tingkatan 4

Tentukan persilangan set-set berikut:

Determine the intersections of the following set:

Contoh /Example : A = { Januari , Oktober }. / A = { January , October }.
A∩B B = { Januari , Mac, Mei, Julai, Ogos, Oktober , Disember}.

B = { January , March, May, July, August, October , December}

A ∩ B = {Januari, Oktober} Pilih unsur yang sama.
Choose the common element.
A ∩ B = {January, October}.

(a) A ∩ C

(b) B ∩ D

(c) A ∩ B ∩ D

UNIT 4

2 Wakilkan persilangan set di bawah dengan menggunakan perwakilan secara simbolik.

Represent the intersection set below by using representation by symbolic.

Diberi / Given

set x = {x: 1 ≤ x ≤ 25, x ialah integer / x is integer}, set P = {nombor ganjil / an odd number},
set Q = {nombor perdana / a prime number}, set R = {gandaan bagi 5 / a multiple of 5}.

Tentukan persilangan set-set berikut: / Determine the intersections of the following sets:

Contoh /Example : P = {1, 3 , 5 , 7 , 9 , 11 , 13 , 15, 17 , 19 , 21, 23 , 25}
P∩Q
Q = {2, 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 ,} Pilih unsur yang sama.
Choose the common element.
P ∩ Q = {3, 5, 7, 11, 13, 17, 19, 23}

(a) P ∩ R P=
R=

P∩R=

(b) Q ∩ R Q=
R=

Q∩R=

(c) P ∩ Q ∩ R P=
Q=

R=

P∩Q∩R=

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Modul • Matematik Tingkatan 4

3 Wakilkan persilangan set di bawah dengan menggunakan perwakilan secara grafik.

Represent the intersection set below by using representation by graphic.

Diberi / Given

set x = {1, 2, 3, 4, 5, 6, 7, 8, 9},
set P = {2, 4, 6, 8},
set Q = {2, 3, 5, 7},
set R = {1, 2, 3}.

Tentukan persilangan set-set berikut:

Determine the intersections of the following sets:

Contoh /Example : (a) P ∩ R
P∩Q

ξ •1

P •3 Q
•8
•2 •5
•4
•6 •7

•9

P ∩ Q = {2}

(b) Q ∩ R (c) P ∩ Q ∩ R UNIT 4

SP4.1.2 Menentukan pelengkap bagi persilangan set.

Determine the complement of the intersection of sets.

NOTA / NOTES

Set pelengkap bagi persilangan dua set A dan B ialah set yang mengandungi semua unsur dalam set
semesta yang bukan unsur dalam persilangan set A dan set B.

The complement of an intersection of two sets, A and B is a set containing all the elements in the universal set which are not elements in

the intersection of set A and set B.

Contoh / Example: B (ii) ξ A (iii) ξ B
(i) ξ B
A
A

(A ∩ B)ʹ (A ∩ B)ʹ C
(A ∩ C)ʹ

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Modul • Matematik Tingkatan 4

4 Tentukan pelengkap bagi set P ∩ Q bagi setiap yang berikut.

Determine the complement of set P ∩ Q for each of the following.

Contoh /Example : (a) x = {5, 6, 7, 8, 9, 10, 11, 12}
P = {gandaan bagi 5 / multiple of 5}
x = {x : 6 ≤ x < 15, x ialah integer} Q = {faktor bagi 15 / factor of 15}

{x : 6 ≤ x < 15, x is an integer}

P = {gandaan 3 / multiple of 3}
Q = {gandaan 4 / multiple of 4}

x = {6, 7, 8, 9, 10, 11, 12, 13, 14} (c) x = {x : 1 ≤ x ≤ 12, x ialah integer / x is an integer}
P = {6, 9, 12} P = {x : x ialah nombor genap}
Q = {8, 12}
{x : x is an even numbers}
(P ∩ Q)' = {6, 7, 8, 9, 10, 11, 13, 14}
Q = {x : x ialah faktor bagi 12}
(b) x = {Tulip, Mawar, Melur, Kenanga}
{x : x is a factor of 12}
{Tulip, Rose, Jasmine, Canangium}

P = {Mawar, Kenanga / Rose, Canangium}
Q = {Tulip, Mawar, Kenanga}

{Tulip, Rose, Canangium}

UNIT 4 5 Cari persilangan set dan pelengkapnya bagi setiap yang berikut.

Find the intersection of sets and its complement for each of the following.

Contoh /Example : (a) ξ •6 •1

ξ •6 •2 P •2
X Y
•3 •1 Q •4
•7 • 11 •5 •9 •3

•4 •9 •8 • 5 • 10
• 10
• 11 •8
•7

X ∩ Y = {1, 11} P∩Q=
(X ∩ Y)' = {2, 3, 4, 5, 6, 7, 8, 9, 10} (P ∩ Q)' =

(b) ξ •9 (c) R

A B •3
•7 •2 •6 • 11 • 13
• 29 • 17
•1 S •7
•4 •3 T •2 •5

•5 • 19
•8
R∩S=
C (R ∩ S)' =
S∩T=
A∩B= (S ∩ T)' =
(A ∩ B)' = R∩S∩T=
(R ∩ S ∩ T)' =
A∩C=
(A ∩ C)' =

B∩C=
(B ∩ C)' =

A∩B∩C=
(A ∩ B ∩ C)' =

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Modul • Matematik Tingkatan 4

SP4.1.3 Menyelesaikan masalah yang melibatkan persilangan set.

Solve problems involving the intersection of sets.

6 Selesaikan setiap yang berikut. / Solve each of the following.

Contoh /Example :

Dalam sebuah kelas yang mengandungi 30 orang murid, didapati 20 orang murid adalah ahli
Persatuan Pengakap, 15 orang murid ahli Kelab Matematik dan 3 orang murid bukan ahli kelab
dan persatuan tersebut. Cari bilangan murid yang menganggotai kedua-dua kelab dan persatuan
tersebut.

In a class containing 30 students, 20 students were members of the Scout Association, 15 students were members of the Mathematics

Club and 3 students were not members for that club and association. Find the number of students who join both club and association.

Biarkan, / Let,
A = bilangan ahli Persatuan Pengakap / number of Scout Association's member, n(A) = 20
B = bilangan ahli Kelab Matematik / the number of Mathematics Club's member, n(B) = 15
x = bilangan murid yang menganggotai kedua-dua kelab dan persatuan tersebut.

the number of students who join both club and association.

ξ B (20 – x) + x + (15 – x) + 3 = 30
A

20 – x x 15 – x 38 – x = 30

3 x=8

Bilangan murid yang menganggotai kedua-dua kelab dan persatuan tersebut ialah 8 orang. UNIT 4

The number of students who join both club and association is 8.

(a) Gambar rajah Venn di bawah menunjukkan (b) Dalam suatu kajian yang melibatkan 60
30 orang pelajar hadir ke sekolah. orang kanak-kanak, didapati bahawa 45
orang kanak-kanak suka memancing, 15
The Venn diagram below shows 30 students attend school. orang kanak-kanak suka berbasikal dan 5
orang kanak-kanak tidak suka kedua-dua
ξ R hobi tersebut. Gunakan gambar rajah Venn
S untuk menghitung bilangan kanak-kanak
yang suka kedua-dua hobi tersebut.
6 45
3 In a survey involving 60 children, it is found that 45 children

32 like fishing, 15 children like cycling and 5 children do not

7 like both hobbies. Use the Venn diagram to calculate the

K number of children who like both hobbies.

x = {Jumlah pelajar yang hadir}

{Number of students who present}

S = {bilangan pelajar yang hadir pada hari
Selasa}

{number of student who present on Tuesday}
R = {bilangan pelajar yang hadir pada hari

Rabu}

{number of student who present on Wednesday}

K = {bilangan pelajar yang hadir pada hari
Khamis}

{number of student who present on Thursday}

Hitung bilangan pelajar yang hadir

Calculate the number of students who present

(i) pada hari Rabu dan Khamis,

on Wednesday and Thursday,

(ii) pada ketiga-tiga hari. / on the three days.

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Modul • Matematik Tingkatan 4

4.2 Kesatuan Set / Union of Sets

NOTA / NOTES

1 Kesatuan set A dan set B adalah semua unsur dalam set A atau set B atau kedua-dua set itu.

The union of set A and set B is the set of common elements in set A or set B or both.

2 Jika A ⊂ B, maka A ∪ B = B.

If A ⊂ B, then A ∪ B = B.

3 Kesatuan set A dan set B ditulis sebagai A ∪ B.

The union of set A and set B is written as A ∪ B.

Contoh / Example:

ξ ξ B
A BA

A∪B A∪B

UNIT 4 SP4.2.1 Menentukan dan menghuraikan kesatuan set menggunakan pelbagai perwakilan.

Determine and describe the union of sets using various representations.

1 Wakilkan kesatuan set di bawah dengan menggunakan perwakilan secara perihalan.

Represent the union set below by using representation by description.

Mei Ling telah mengelaskan set nombor perdana seperti di bawah:

Mei Ling has classified the set of prime number as below:

Set x ialah set nombor perdana kurang daripada 20. / Set ξ is the set of prime number less than 20.
Set P ialah nombor ganjil. / Set P is an odd number.
Set Q ialah gandaan 3. / Set Q is a multiple of 3.
Set R ialah faktor bagi 15. / Set R is a factor of 15.

Tentukan kesatuan set-set berikut: / Determine the union of the following sets:

Contoh /Example : (a) P ∪ R

P∪Q
P = {3, 5, 7, 11, 13, 17, 19}.
Q = {3}.

P ∪ Q = {1, 3, 5, 7, 11, 13, 17, 19}.


Gabungan sama unsur kedua-dua set.
Combine all the elements in both set.

(b) Q ∪ R (c) P ∪ Q ∪ R

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Modul • Matematik Tingkatan 4

2 Wakilkan kesatuan set di bawah dengan menggunakan perwakilan secara simbolik.

Represent the union set below by using representation by symbolic.

Diberi / Given
set x = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9},
set A = {1, 2, 3, 4},
set B = {0, 2, 4, 6, 8},
set C = {0, 3, 6, 9}.

Tentukan kesatuan set-set berikut:

Determine the union of the following sets:

Contoh /Example : (a) A ∪ C =
A ∪ B = {0, 1, 2, 3, 4, 6, 8}

(b) B ∪ C = (c) A ∪ B ∪ C =

3 Wakilkan kesatuan set di bawah dengan menggunakan perwakilan secara grafik. UNIT 4

Represent the union set below by using representation by graphic.

Diberi / Given
set x = {integer di antara 5 hingga 15 / integer between 5 until 15}
set X = {12, 15}
set Y = {nombor perdana / a prime number}
set Z = {gandaan bagi 3 / a multiple of 3}

Tentukan kesatuan set-set berikut: (a) X ∪ Z =

Determine the union of the following sets:

Contoh /Example :

X ∪ Y = {5, 7, 11, 12, 13, 15}

ξ • 15 • 11 Y
X • 12 • 7 • 13

•5

(b) Y ∪ Z = (c) X ∪ Y ∪ Z =

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Modul • Matematik Tingkatan 4

SP4.2.2 Menentukan pelengkap bagi kesatuan set.

Determine the complement of the union of sets.

Set pelengkap bagi kesatuan dua set, A dan B ialah set yang ξ B
mengandungi semua unsur dalam set semesta (x) yang bukan A (A ∪ B)ʹ
unsur set A atau set B. Set (A ∪ B)' diwakili dengan kawasan yang
berlorek dalam gambar rajah Venn di sebelah.

The complement of the union of two sets, A and B is a set containing all the elements in the

universal set (x) which are not elements of set A or set B. Set (A ∪ B)' is represented by the
shaded region in the Venn diagram on the right.

4 Tentukan pelengkap bagi set A ∪ B bagi setiap yang berikut.

Determine the complement of set A ∪ B for each of the following.

Contoh /Example : (a) x = {x : 3 ≤ x < 10, x ialah integer / x is an integer}
A = {gandaan 2 / multiple of 2}
x = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} B = {gandaan 4 / multiple of 4}
A = {gandaan bagi 3 / multiple of 3}
B = {faktor bagi 12 / factor of 12} (A ∪ B)' =

(A ∪ B)' = {5, 7, 8, 10, 11, 13}

UNIT 4 (b) x = {x : 1 ≤ x ≤ 12, x ialah integer / x is an integer} (c) x = {f, g, h, i, j, k, l, m, n}
A = {x : x ialah nombor perdana} A = {h, k, l, m}
B = {k, m}
{x : x is a prime number}
(A ∪ B)' =
B = {x : x ialah faktor perdana bagi 12}

{x : x is a prime factor of 12}

(A ∪ B)' =

5 Cari pelengkap bagi setiap kesatuan yang berikut. / Find the complement of each of the following unions of sets.

Contoh /Example : (a) ξ •a

ξ •6 •2 P •b •e
X Y Q •d •f
•3 •1
• 7 •11 •5 •c •i •k

•4 •9 •8 •h •g
• 10 •j

(X ∪ Y)' = {2, 4, 6, 9, 10} (P ∪ Q)' =

(b) ξ •9 (c)

AB R •3 T
•1 •2 •5 • 11

•3 • 29 • 17 • 13
•8 •7 S •2 •7 •5

•6
•4

C

(A ∪ B)' = • 19
(A ∪ C)' =
(B ∪ C)' = (R ∪ S)' =
(A ∪ B ∪ C)' = (S ∪ T)' =
(R ∪ S ∪ T)' =

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Modul • Matematik Tingkatan 4

SP4.2.3 Menyelesaikan masalah yang melibatkan kesatuan set.

Solve problems involving the union of sets.

6 Selesaikan setiap yang berikut. / Solve each of the following.

Contoh /Example :

Keputusan ujian pertengahan tahun kelas 5 Amal bagi mata pelajaran Matematik dan Sains adalah
seperti berikut: / The examination results of the mid year test of 5 Amal for Mathematics and Science subjects are as follows:

20 orang pelajar lulus Sains. / 20 students pass Science.
15 orang pelajar lulus Matematik. / 15 students pass Mathematics.
10 orang pelajar lulus kedua-dua subjek tersebut. / 10 students pass both subjects.

Cari bilangan pelajar yang lulus Sains atau Matematik atau kedua-dua subjek tersebut.

Find the number of students who pass Science or Mathematics or both subjects.

Biarkan, / Let,
x = {bilangan murid kelas 5 Amal / number of students in class 5 Amal}
A = {bilangan murid yang lulus Sains / number of students who pass Science}, n(A) = 20.
B = {bilangan murid yang lulus Matematik / number of students who pass Mathematics}, n(B) = 15.
A ∩ B = {bilangan murid yang lulus kedua-dua subjek / number of students who pass both subjects}

ξ A B Bilangan pelajar yang lulus Sains atau Matematik atau kedua-dua subjek
ialah
20 – 10 10 15 – 10 UNIT 4
The number of students who pass Science or Mathematics or both subjects are

(20 – 10) + 10 + (15 – 10) = 25 orang pelajar / students

(a) Hasil kajian ke atas 60 buah rumah di suatu kawasan perumahan menunjukkan bahawa
48 buah rumah memiliki siaran Astro Ria dan 20 buah rumah memiliki siaran Astro Njoi. Hanya
12 buah rumah sahaja memiliki kedua-dua siaran tersebut.

In a survey on 60 houses in a residential area shows that there are 48 houses have Astro Ria broadcast and 20 houses have Astro
Njoi broadcast. Only 12 houses have both broadcasts.

(i) Lukis gambar rajah Venn untuk mewakili maklumat tersebut.

Draw a Venn diagram to represent the information.

(ii) Cari bilangan rumah yang mempunyai siaran tersebut.

Find the number of houses that have those broadcasts.

(b) Dalam satu rombongan yang terdiri daripada 48 orang pelajar, didapati 30 orang pelajar
memakai cermin mata, 20 orang lagi didapati memakai jam tangan dan 10 orang pelajar
memakai kedua-dua aksesori.

In a trip of 48 students, 30 students wear glasses, other 20 students wear watches and 10 students wear both of the accesories.

Hitung / Calculate
(i) bilangan pelajar yang tidak memakai kedua-dua aksesori,

the number of students who do not wear both of the accessories,

(ii) bilangan pelajar yang memakai cermin mata atau jam tangan atau kedua-dua aksesori.

the number of students wear glasses or watches or both of the accessories.

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Modul • Matematik Tingkatan 4

4.3 Gabungan Operasi Set / Combined Operations on Sets

NOTA / NOTES

Gabungan operasi set adalah gabungan operasi di dalam persilangan dan kesatuan ataupun
kedua-duanya sekali termasuk pelengkap di dalam set tersebut. Gabungan operasi set perlu
diselesaikan dari kiri ke kanan, namun jika terdapat operasi dalam kurungan, operasi dalam
kurungan mesti diselesaikan dahulu.

Combined operations on sets is a combination of operations in intersection and union or both of which are including complement

in the set. Combination of set operation needs to be solved from left to right, but if there are operations in bracket, the operations in

brackets must be carried out first.

Contoh / Example: (ii) (A ∩ B) ∪ C
(i) A ∪ B ∩ C

dan sebagainya. / and so on.

SP4.3.1 Menentukan dan menghuraikan gabungan operasi set menggunakan pelbagai perwakilan.

Determine and describe the combined operations on sets using various representations.

1 Selesaikan yang berikut.

Solve the following.

UNIT 4 Contoh /Example : (a) Diberi / Given
x = {a, b, c, d, e, f, g, h, i}
Diberi / Given A = {a, c, d, f, g, i}
x = {x : 1 ≤ x ≤ 10, x ialah integer / x is an integer} B = {a, b, c, d}
A = {1, 2, 3, 4, 5, 6} C = {a, h}
B = {2, 4, 6, 8, 10}
C = {3, 4, 5, 6, 7} Cari / Find
(i) A ∩ (B ∪ C),
(i) Senaraikan semua unsur bagi A ∪ B ∩ C.
(ii) n [A ∩ (B ∪ C)].
List all the elements of A ∪ B ∩ C.

(ii) Cari / Find n(A ∪ B ∩ C).

A ∪ B = {1, 2, 3 , 4 , 5 , 6 , 8, 10}
C = { 3 , 4 , 5 , 6 , 7}

A ∪ B ∩ C = {3, 4, 5, 6}
n(A ∪ B ∩ C) = 4

(b) Diberi / Given (c) P = {x : 3 ≤ x < 13, x ialah nombor genap}
x = {x : x ialah nombor bulat kurang
daripada 11} {x : 3 ≤ x <13, x is an even number}

{x : x is a whole number which are less than 11} Q = {x : 3 ≤ x < 13, x ialah gandaan bagi 3}

A = {nombor genap / even number} {x : 3 ≤ x < 13, x is multiple of 3}
B = {faktor bagi 42 / factor of 42}
C = {faktor sepunya bagi 18 dan 36} R = {x : 3 ≤ x < 13, x ialah nombor perdana}

{the common factor of 18 and 36} {x : 3 ≤ x < 13, x is a prime number}

Cari / Find Cari / Find
(i) (A ∪ B) ∩ C,   (ii)  n[(A ∪ B) ∩ C]. (i) (P ∩ Q) ∪ R,
(ii) n[(P ∩ Q) ∪ R].

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Modul • Matematik Tingkatan 4

2 Lorekkan rantau yang mewakili operasi bagi set yang berikut.

Shade the region which represents the operations of set of each of the following.

Contoh /Example : B (a) (A ∪ B) ∩ C B (b) (A ∩ B) ∪ C
A∩B∪C
A A
A C

B

C
C

SP4.3.2 Menentukan pelengkap bagi gabungan operasi set.

Determine the complement of combined operations on sets.

3 Selesaikan setiap yang berikut.

Solve each of the following.

Contoh /Example : (a) x = {x : 30 < x < 38, x ialah integer} UNIT 4

x = { x : x ialah nombor perdana kurang {x : 30 < x < 38, x is an integer}
daripada 20 / x : x is a prime number less than 20}
A = {x : x ialah nombor ganjil / x : x is an odd number} M = {x : x ialah nombor perdana}
B = {x : x ialah gandaan bagi 3 / x : x is a multiple of 3}
C = {x : x ialah faktor bagi 20 / x : x is a factor of 20} {x : x is a prime number}

(i) A ∩ (B ∪ C)'  (ii) n[A ∩ (B ∪ C )'] N = {x : x mempunyai dua digit yang sama}

(i) B ∪ C = {2, 3, 5} {x : x has two same digits}
(B ∪ C)' = { 7 ,11,13,17,19}
A = {3, 5, 7 ,11,13,17,19} P = {x : x ialah faktor bagi 37}

{x : x is factor of 37}

Cari / Find

(i) N ∪ (M ∩ P)',  (ii)  n[N ∪ (M ∩ P)'].

A ∩ (B ∪ C)' = {7, 11, 13, 17, 19}

(ii) n[A ∩ (B ∪ C)'] = 5

(b) x = R ∪ S ∪ T (c) Gambar rajah Venn menunjukkan bilangan
R = {1, 3, 5, 7, 9} unsur dalam set x , set P, set Q dan set R.
S = {2, 3, 5, 7, 9}
T = {1, 5, 10} The Venn diagram shows the number of elements in set x ,

Cari / Find set P, set Q and set R.
(i) S ∩ (R ∪ T)'
(ii) n[S ∩ (R ∪ T)'] ξ R Q
P 34
2
1 5

Cari / Find
(i) R ∪ (P ∩ Q)', (ii) n[R ∪ (P ∩ Q)'].



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Modul • Matematik Tingkatan 4

SP4.3.3 Menyelesaikan masalah yang melibatkan gabungan operasi set.

Solve problems involving combined operations on sets.

4 Selesaikan setiap yang berikut.

Solve each of the following.

Contoh /Example :
Hasil daripada tinjauan 43 orang pekerja di sebuah Syarikat Pembinaan Maju menunjukkan
bahawa 24 orang pekerja memiliki kereta, 19 orang pekerja memiliki motosikal dan 16 orang
pekerja memiliki basikal. Daripada tinjauan itu juga, 12 orang pekerja memiliki kereta sahaja,
8 orang pekerja memiliki motosikal sahaja, 7 orang pekerja memiliki kereta dan motosikal sahaja
dan 3 orang pekerja memiliki kereta dan basikal sahaja.

A result of a survey of 43 employees in Pembinaan Maju Company shows that 24 employees have a car, 19 employees have a

motorcycle and 16 employees have a bicycle. From the survey also, 12 employees have a car only, 8 employees have a motorcycle only,

7 employees have a car and motorcycle only and 3 employees have a car and bicycle only.

UNIT 4 Cari

Find

(i) bilangan pekerja yang memiliki kereta, motosikal dan basikal,

the number of employees who have a car, motorcycle and bicycle,

(ii) bilangan pekerja yang memiliki motosikal dan basikal sahaja,

the number of employees who have a motorcycle and bicycle only,

(iii) bilangan pekerja yang memiliki basikal sahaja. / the number of employees who have a bicycle only.

Biarkan / Let ξ B
x = {Bilangan pekerja yang memiliki kenderaan} A

{The number of employee who have vehicles} 12 78
3 x
A = {Bilangan pekerja yang memiliki kereta}
y
{The number of employee who have a car}
z
B = {Bilangan pekerja yang memiliki motosikal} C

{The number of employee who have a motorcycle}

C = {Bilangan pekerja yang memiliki basikal}

{The number of employee who have a bicycle}

(i) Diberi / Given n(A) = 24
12 + 3 + x + 7 = 24
x = 2

Jadi, 2 orang pekerja memiliki kereta, motosikal dan basikal.

Hence, 2 employees have car, motorcycle and bicycle.

(ii) Diberi / Given, n(B) = 19
7 + 2 + y + 8 = 19
y = 2

Jadi, 2 orang pekerja memiliki motosikal dan basikal sahaja.

Hence, 2 employees have motorcycle and bicycle only.

(iii) Diberi / Given, n(C) = 16
3 + 2 + 2 + z = 16
z = 9

Jadi, 9 orang pekerja memiliki basikal sahaja.

Hence, 9 employees have a bicycle only.

© Nilam Publication Sdn. Bhd. 72

Modul • Matematik Tingkatan 4 UNIT 4

(a) Keputusan ujian bulanan bagi 80 orang murid dalam mata pelajaran Matematik, Matematik
Tambahan dan Sains adalah seperti berikut:

The monthly test results of 80 students in the Mathematics, Additional Mathematics and Science subjects are as follows:

18 orang pelajar lulus Matematik sahaja.

18 students pass Mathematics only.

17 orang pelajar lulus Matematik Tambahan sahaja.

17 students pass Additional Mathematics only.

21 orang pelajar lulus Sains sahaja.

21 students pass Science only.

12 orang pelajar lulus Matematik dan Matematik Tambahan.

12 students pass Mathematics and Additional Mathematics.

13 orang pelajar lulus Matematik dan Sains.

13 students pass Mathematics and Science.

11 orang pelajar lulus Matematik Tambahan dan Sains.

11 students pass Additional Mathematics and Science.

Tiada pelajar yang gagal dalam ketiga-tiga mata pelajaran tersebut.

No student fails in the three subjects.

Cari / Find
(i) bilangan pelajar yang lulus dalam ketiga-tiga mata pelajaran,

the number of students who passes the three subjects,

(ii) bilangan pelajar yang lulus dua mata pelajaran sahaja,

the number of students who passes only two subjects,

(iii) bilangan pelajar yang lulus mata pelajaran Sains.

the number of students who passes the Science subject.

73 © Nilam Publication Sdn. Bhd.

Modul • Matematik Tingkatan 4

(b) Dalam sebuah kelas terdapat 34 orang pelajar lelaki. Setiap murid lelaki bermain sekurang-
kurangnya satu daripada permainan iaitu badminton(B), bola sepak(F) dan hoki(H). Diberi
bahawa bilangan murid lelaki yang bermain bola sepak adalah dua kali bilangan murid lelaki
yang bermain badminton dan hoki.

There are 34 boys in a class. Each of the boys plays at least one of the game that is badminton(B), football(F) and hockey(H).

Given that the number of boys plays football is twice the number of boys plays badminton and hockey.

Cari / Find BF
(i) bilangan murid lelaki yang bermain badminton sahaja,
y 27
the number of boys who plays badminton only, x 4

(ii) bilangan murid lelaki yang bermain badminton dan hoki sahaja, 3

the number of boys who plays badminton and hockey only, 5

(iii) bilangan murid lelaki yang bermain satu permainan sahaja.

the number of boys who plays only one game.

H

UNIT 4

(c) Sebuah syarikat televisyen menjalankan kaji selidik terhadap 150 orang mengenai filem
kegemaran mereka. Data yang diperoleh adalah seperti berikut:

A television company carried out a survey on 150 people about their favourite movies. The data obtained were as follow:

67 orang suka menonton filem Inggeris.

67 people preferred watching English movies.

78 orang suka menonton filem Melayu.

78 people preferred watching Malay movies.

75 orang suka menonton filem Korea.

75 people preferred watching Korean movies.

25 orang suka menonton filem Korea dan Melayu.

25 people preferred watching Korean and Malay movies.

20 orang suka menonton filem Melayu dan Inggeris.

20 people preferred watching Malay and English movies.

5 orang suka menonton ketiga-tiga filem tersebut.

5 people preferred all three types of movie.

4 orang tidak suka menonton ketiga-tiga filem tersebut.

4 people did not like any of three types of movie.

© Nilam Publication Sdn. Bhd. 74

Modul • Matematik Tingkatan 4

Cari / Find
(i) bilangan orang yang suka menonton filem Melayu sahaja,

the number of people who preferred watching Malay movies only,

(ii) bilangan orang yang suka menonton filem Melayu dan Korea tetapi bukan filem Inggeris,

the number of people who preferred watching Malay and Korean movies but not English movies,

(iii) bilangan orang yang suka menonton filem Melayu atau Inggeris tetapi bukan filem Korea.

the number of people who preferred watching Malay or English movies but not Korean movies.

UNIT 4

LATIHAN PENGUKUHAN / ENRICHMENT EXERCISE

Kertas 1 /Paper 1 Cari n(G ∩ H)'. C 8
D 9
1 Rajah di bawah menunjukkan bilangan unsur Find n(G ∩ H)'.
dalam set P, set Q and set R. Diberi set semesta,
ξ = P ∪ Q ∪ R dan n(ξ) = 28. A 6
B 7
Diagram below shows the number of elements in set P, set Q
and set R. Given universal set, ξ = P ∪ Q ∪ R and n(ξ) = 28. 3 3 Rajah di bawah menunjukkan gambar rajah

Q Venn dengan set semesta x = E ∪ F ∪ G.
PR Diagram below shows the Venn diagram with the universal set

7 4 3 3x x – 2

x = E ∪ F ∪ G.  3

Nilai bagi x ialah / The value of x is ξ F
G

A 6 C 4 E

B 5 D 3

2 Diberi set semesta, Nyatakan set yang mewakili rantau berlorek.

Given the universal set,   3 State the set which represents the shaded region.

x = {x : 2 ≤ x ≤ 9, x ialah nombor bulat} A G ∪ (F ∩ E)
{x : 2 ≤ x ≤ 9, x is a whole number} B G ∪ (F ∩ E')
G = {x : x ialah nombor perdana} C G ∪ (F' ∩ E)
D G ∪ (F ∩ E)'
{x : x is a prime number}
75 © Nilam Publication Sdn. Bhd.
H = {x : x ialah nombor gandaan 3}

{x : x is a multiple of 3}

Modul • Matematik Tingkatan 4

4 Diberi / Given x = R ∪ S ∪ T, 8 Gambar rajah Venn menunjukkan bilangan

Set R = {2, 3, 4, 5, 6}, unsur dalam set R, set S dan set T.

Set S = {6, 7, 8, 9}, The Venn diagram below shows the number of elements in set

Set T = {4, 5}. R, set S and set T.   3

Senaraikan unsur bagi (R ∪ T) ∩ S. R S
List the elements of (R ∪ T) ∩ S.  4 27
3 5 38

A {6} C {6, 7, 8, 9} 6

B {4, 5} D {2, 3, 4, 5, 6}

5 Rajah di bawah menunjukkan hubungan T

di antara set ξ, set R dan set S. n(S ∩ T ∪ R) ialah / is
Diagram below shows the relationship between set ξ, set R and A 14
B 19
set S.  3 C 20
D 22
ξ S
R

Diberi n(ξ) = 35, n(R ∩ S) = 5, n(R) = 17 dan 9 Gambar rajah Venn di bawah menunjukkan
n(R ∪ S)' = 9. Hitung bilangan unsur dalam
rantau berlorek. bilangan unsur dalam set P, set Q and set R.

Given n(ξ) = 35, n(R ∩ S) = 5, n(R) = 17 and n(R ∪ S)' = 9. The Venn diagram below shows the number of elements in
Calculate the number of elements in the shaded region. 
set P, set Q and set R.   3

UNIT 4 A 9 C 19 ξ Q R
P 9

B 14 D 24 5 20

6 Rajah di bawah menunjukkan set P, set Q Diberi bahawa x = P ∪ Q ∪ R dan n(x) = 40.
Cari nilai R ∪ (P ∩ Q).
dan set R dengan keadaan set semesta
It is given that x = P ∪ Q ∪ R and n(x) = 40. Find the value of
x = P ∪ Q ∪ R. R ∪ (P ∩ Q).
Diagram below shows set P, set Q and set R such that the
A 6
universal set x = P ∪ Q ∪ R.  3
B 7
P y R
Q z C 8

wx D 9

Antara rantau w, x, y atau z, yang manakah 10 Rajah di bawah ialah ambar rajah Venn dengan
mewakili (Q ∪ P') ∩ R?
set semesta x = X ∪ Y.
Which of the region w, x, y or z, represent (Q ∪ P') ∩ R? Diagram below is a Venn diagram with the universal

A w C y set x = X ∪ Y.  3

B x D z XY

7 Diberi set R = {2, 3, 5, 6}, S = {3, 6, 9, 12} dan

T = {2, 3, 9, 13}. Set R ∩ (S ∪ T) ialah
Given set R = {2, 3, 5, 6}, S = {3, 6, 9, 12} and T = {2, 3, 9, 13}.
Diberi n(x) = 58, n(X) = 40, n(X ∩ Y) = 8 dan
Set R ∩ (S ∪ T) is   3 n(X') = p + 7. Cari nilai p.

A {2, 3, 6} Given n(x) = 58, n(X) = 40, n(X ∩ Y) = 8 and n(X') = p + 7. Find
the value of p.
B {2, 3, 5, 6}

C {3, 6, 9, 12} A 2 C 9

D {2, 3, 6, 9, 12} B 5 D 11

© Nilam Publication Sdn. Bhd. 76

Modul • Matematik Tingkatan 4

Kertas 2 /Paper 2

1 Jadual di bawah menunjukkan data yang diperoleh daripada satu kaji selidik ke atas 80 orang pelajar
yang mendaftar masuk tiga kelas yang berbeza. Cari

The table below shows the data obtained from a survey of 80 students who register in three different classes. Find  4

(a) bilangan pelajar yang mendaftar masuk ketiga- Kelas Bilangan pelajar
tiga kelas tersebut,
Class Number of students
the number of students that register in the three classes, 
Nyanyian 48
(b) bilangan pelajar yang mendaftar masuk kelas
nyanyian atau muzik dan juga tarian, Singing 32

the number of students that register in singing or music and Tarian

dance classes,  Dancing

(c) bilangan pelajar yang mendaftar masuk kelas Nyanyian dan muzik
nyanyian sahaja atau muzik sahaja atau tarian 15
sahaja
sahaja.
Singing and music only

the number of students that register in singing or music or dance Tarian dan nyanyian 9
class only.  sahaja

Jawapan / Answer: Dancing and singing only

Nyanyian sahaja 18

Singing only

Tarian sahaja 11

Dancing only

UNIT 4

2 Sebuah pusat pengajian tinggi menyediakan tiga jenis bahasa asing seperti F K

bahasa Perancis, bahasa Jepun dan bahasa Arab. Gambar rajah Venn di bawah 6 79

menunjukkan bilangan pelajar yang mempelajari tiga bahasa asing dalam p
sekumpulan 40 orang pelajar. p4

A higher education center provides three different types of foreign languages ​l​ike French, Japanese and q

Arabic language. The Venn diagram below shows the number of students who study three foreign languages​ J

in a group of 40 students.  4

Diberi bahawa x = F ∪ J ∪ K, / Given that x = F ∪ J ∪ K,
set F = {bilangan pelajar yang mempelajari bahasa Perancis / the number of students who study France language}
set J = {bilangan pelajar yang mempelajari bahasa Jepun / the number of students who study Japanese language}
set K = {bilangan pelajar yang mempelajari bahasa Arab / the number of students who study Arabic language}

Bilangan pelajar yang mempelajari bahasa Arab ialah 23 orang. Cari

The number of students who study Arabic language is 23. Find

(a) nilai bagi p dan q, / the value of p and q, 

(b) bilangan pelajar yang mempelajari satu bahasa asing sahaja,

the number of students who study only one foreign language, 

(c) bilangan pelajar yang mempelajari bahasa Jepun.

the number of students who study Japanese language 

77 © Nilam Publication Sdn. Bhd.

Modul • Matematik Tingkatan 4

Jawapan / Answer:

3 (a) Diberi bahawa / Given that   4
set W = {nombor perdana / a prime number} dan / and set V = {2, 3, 5}.

Lengkapkan gambar rajah Venn di ruang jawapan untuk menunjukkan hubungan antara set W
dan set V. / Complete a Venn diagram in the answer space to show a relationship between set W and set V. 

(b) Gambar rajah Venn di sebelah menunjukkan set P, set Q dan set P QR
R. Set semesta x = P ∪ Q ∪ R.

The Venn diagram in the right shows set P, set Q and set R. Universal set x = P ∪ Q ∪ R.

Nyatakan hubungan yang diwakili oleh gambar rajah Venn di
sebelah. / State the relationship represented by the Venn diagram above. 

UNIT 4 Jawapan / Answer:

(a) (b)

W
V

Aktiviti PAK-21 / 21st Century Activities

Operasi Set / Set Operation
1 Murid dibahagikan dalam beberapa kumpulan. Setiap kumpulan terdiri daripada 4 atau 5 orang

pelajar.

Students are divided into a few group. Each group consists of 4 or 5 students.

2 Guru menyediakan kajian kes berkaitan Operasi Set.

Teacher provides case studies related to Set Operation.

3 Setiap kumpulan dikehendaki menyediakan buku skrap dan hasil akan dibentangkan di dalam
kelas.

Each group is required to provide a scrap book and the results will be presented in the class.

4 Tugasan terbaik akan dipamerkan sewaktu pertandingan STEM.

The best assignments will be shown during the STEM competition.

Kajian Kes / Case study

Buat rujukan di perpustakaan sekolah anda, melalui internat atau sumber lain yang bersesuaian.
Dapatkan maklumat berkaitan kajian kes, contohnya jenis-jenis darah, jenis kategori yang
dipertandingkan dalam sukan dan topik berkaitan dengan STEM. Kelaskan kajian anda kepada set-
set tertentu dan wakilkan dengan menggunakan gambar rajah Venn. Kaitkan masalah kajian kes di
atas dalam pelbagai bentuk operasi set.

Do a reference in your school library, through the internet or other appropriate sources. Get information on case studies, such as blood
types, sports related categories and STEM related topics. Classified your study into specific sets and represent them using Venn diagrams.
Relate the case study problems above to the different types of set operation.

© Nilam Publication Sdn. Bhd. 78