UNIVERSITI MALAYSIA SARAWAK KOTA SAMARAHAN SARAWAK FIFTH, SIXTH, EIGHTH & NINTH SEMESTER EXAMINATION, 2013 SESSION MTH2033 DISCRETE MATHEMATICS

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1 UNIVERSITI MALAYSIA SARAWAK KOTA SAMARAHAN SARAWAK FAKULTI SAINS KOMPUTER & TEKNOLOGI MAKLUMAT (Faculty of Computer Science & Information Technology) Diploma in Multimedia Technology FIFTH, SIXTH, EIGHTH & NINTH SEMESTER EXAMINATION, 2013 SESSION DMTW-E-F 1/11, DMTW-E-F 2/11, DMTQ-E-F 1/10, DMTQ-E-F 2/10, DMTQ-E-F 1/11 MTH2033 DISCRETE MATHEMATICS BAIDA SHIHA BAHARUDDIN, AZIZI BIN AHMADI 10 APRIL PM 4.30 PM (2 ½ JAM/ HOURS) INSTRUCTION TO STUDENT 1. This Examination paper has THIRTEEN (13) printed pages. 2. This question paper consists of THREE (3) sections. Section A & B : Answer ALL questions. Section C : Answer FOUR (4) questions ONLY. 3. Please write all your answers in the Answer Booklet provided.

2 BAHAGIAN A SECTION A : SOALAN PELBAGAI PILIHAN (10 MARKAH) : MULTIPLE CHOICE QUESTIONS (10 MARKS) ARAHAN INSTRUCTIONS : Jawab SEMUA soalan : Answer ALL questions 1. Kenyataan manakah BUKAN satu usulan? / Which of the statements is NOT a proposition? A. Kangar adalah ibu negeri Perlis / Kangar is the capital of Perlis. B. Masukkan buah ini ke dalam bakul / Put this fruit into the basket. C. 5 > 6. D. Bulan diperbuat daripada keju hijau / The moon is made of green cheese. 2. (p q) adalah bersamaan. / (p q) is equivalence to. A. p q. B. p q. C. p q. D. p q. 3. Andaikan set A = {a, b, c, d} dan set B = {a, b, c, d, e, f, g, h}. Yang manakah perkaitan di bawah yang menunjukkan hubungan set A dan B? Let A = {a, b, c, d} and set B = {a, b, c, d, e, f, g, h}. Which of the following best describe the relationship between set A and set B? A. A B. B. A B. C. A B. D. A B. BSB/AA Page 2 of 13

3 4. Tentukan nilai bagi ¾ + 1/2. / Determine the value of ¾ + 1/2. A. 0. B. 5/2. C. 1. D. 3/4. 5. Apakah hasil bahagi dan baki bagi 3-11? / What are the quotient and remainder for 3-11? A. q = - 4 & r = 1. B. q = -3 & r = - 2. C. q = - 4 & r = - 2. D. Tiada jawapan / No answer = 8 A. 1F. B. 31. C. 37. D Cari P(8, 5):/ Find P(8, 5): A. 56. B. 40. C D BSB/AA Page 3 of 13

4 8. Cari C(8, 5): / Find C(8, 5): A. 56. B. 40. C D Apakah kebarangkalian bagi nombor integer yang dipilih daripada 30 nombor positif integer yang pertama adalah nombor perdana? / What is the probability that selected integer chosen from first 30 positive integer is prime numbers? A. 9/30. B. 10/30. C. 11/30. D. 12/ Apakah kebarangkalian hari dipilih secara rawak di dalam setahun (andaikan ianya tahun lompat) adalah di dalam bulan Februari?/ What is the probability that a randomly selected day of the year (assume it is a leap year) is in February? A. 29/366. B. 28/366. C. 29/365. D. 28/365. BSB/AA Page 4 of 13

5 BAHAGIAN B SECTION B : SOALAN PENDEK (50 MARKAH) : SHORT QUESTIONS (50 MARKS) ARAHAN INSTRUCTIONS : Jawab SEMUA soalan : Answer ALL questions Soalan / Question 1 Diberi bahawa p: Hari ini hari selasa, q: Hari ini hujan dan r: hari ini sejuk. Tuliskan setiap ayat yang berikut menggunakan p, q dan r dan juga penyambung logik. Given that p: today is Tuesday, q: Today is Raining and r: today is cold. Write each of the following in terms of p, q, r and logical connectives. a) Sekiranya hari ini hujan dan sejuk, jadi hari ini hari Selasa. / If today is raining and cold, then today is Tuesday. b) Hari ini sejuk jika dan hanya jika hari ini hujan. / Today is cold if and only if today raining. (5 markah/mark) Soalan / Question 2 Sekiranya A = {a, b, c, d} dan B = {1, 2, 3}. Tentukan samada hubungan R dari ke B adalah sebuah fungsi. Sekiranya ai adalah fungsi, berikan jalatnya. Let A = {a, b, c, d} and B = {1, 2, 3}. Determine whether the relation R from A to B is a function. If it is a function, give a range. a) R = { (a,b), ( b,2 ), ( c,1 ) }. b) R = { ( a,1 ), ( b,1 ) (c,1 ),( d,1 ) }. (5 markah/mark) BSB/AA Page 5 of 13

6 Soalan / Question 3 Apakah nilai bagi hasil tambah berikut? / What are the values of these sums? a) 2 3. b) (2i k ). c) k=0 i=0. (5 markah/mark) Soalan / Question 4 Carikan faktorrisasi perdana bagi setiap yang berikut: / Find the prime factorization for each the following: a) 300. b) 452. (5 markah/mark) Soalan / Question 5 Andaikan B(x) adalah kenyataan x mempunyai seekor burung, C(x) adalah kenyataan x mempunyai seekor kucing, dan H(x) adalah kenyataan x mempunyai seekor tikus. Nyatakan setiap kenyataan yang berikut di dalam bentuk B(x), C(x), H(x), pengkuantiti, dan sambungan logikal. Andaikan domain mengandungi semua pelajar di dalam kelas anda. Let B(x) be the statement x has a bird, let C(x) be the statement x has a cat, and let H(x) be the statement x has a hamster. Express each of these statements in terms of B(x), C(x), H(x), quantifiers, and logical connectives. Let the domain consist of all students in your class. BSB/AA Page 6 of 13

7 a) Seorang pelajar di dalam kelas anda mempunyai seekor burung, seekor kucing dan seekor tikus. / A student in your class has a bird, a cat and a hamster. (1 markah/mark) b) Semua pelajar di dalam kelas anda mempunyai seekor burung, seekor kucing dan seekor tikus. / All students in your class have a bird, a cat, and a hamster. (1 markah/mark) c) Sesetengah pelajar di dalam kelas anda mempunyai seekor burung dan seekor kucing tetapi tidak seekor tikus. Some students in your class have a bird and a cat but not a hamster. (2 markah/marks) d) Tiada pelajar di dalam kelas anda mempunyai seekor burung, seekor kucing atau seekor tikus. / No student in your class has a bird, a cat or a hamster. (1 markah/marks) Soalan / Question 6 Diberi set A = {1, 2, 3, 4}, set B = {2, 4, 6, 8} and set C = {3, 4, 5, 6}. Selesaikan operasi set yang berikut: / Given set A = {1, 2, 3, 4} and set B = {2, 4, 6, 8} and set C = {3, 4, 5, 6}. Solve the following set operations. a) (A B) C (1 markah/marks) b) A (B C) (1 markah/marks) BSB/AA Page 7 of 13

8 c) A B C (1 markah/marks) d) (A B) (A C) (2 markah/marks) Soalan / Question 7 Selesaikan yang berikut: / Solve the followings: a) Cari hasilbahagi (q) dan baki (r) untuk / Find the quotient (q) and remainder (r) or (2 markah/marks) b) Cari gcd dan lcm untuk dan / Find gcd and lcm for and (3 markah/marks) Soalan / Question 8 a) Cari bilangan plat kereta yang boleh dibuat jika setiap plat mengandungi dua huruf yang berbeza diikuti oleh tiga digit yang berbeza / Find the number of license plates can be made is each plate contains two distinct letters followed by three distinct digits. (2markah/marks) b) Selesaikan masalah di dalam soalan 8(a) jika digit yang pertama tidak boleh sifar (0) / Solve the problem in question 8(a) if the first digit cannot be zero (0). (3 markah/marks) BSB/AA Page 8 of 13

9 Soalan / Question 9 Dengan menggunakan algorithma Euclidean, cari gcd (12345, 54321): / By using Euclidean algorithm, find gcd (12345, 54321): (5 markah / marks) Soalan / Question 10 Dengan menggunakan Teorem Binomial, apakah kembangan (x + y) 4. / By using Binomial Theorem, what is the expansion of (x + y) 4. (5 markah / marks) BSB/AA Page 9 of 13

10 BAHAGIAN C SECTION C : SOALAN PANJANG (40 MARKAH) : LONG QUESTIONS (40 MARKS) ARAHAN INSTRUCTIONS : Jawab EMPAT soalan sahaja : Answer FOUR questions only Soalan / Question 1 Tunjukkan bahawa implikasi yang berikut adalah satu tautology dengan menggunakan Jadual Kebenaran / Show that the following implication is a tautology by using truth tables. [(p q) (p r) (q r)] r (10 markah / marks) Soalan / Question 2 Tunjukkan bahawa hipotesis Ianya tidak panas tengahari ini dan ianya lebih sejuk daripada semalam. Kami boleh pergi berenang jika hanya ianya panas. Jika kami tidak pergi berenang, maka kami akan mengambil perjalanan kolek dan Jika kami mengambil perjalanan kolek, maka kami akan pulang pada waktu senja menjurus kepada kesimpulan Kami akan pulang pada waktu senja. Show that the hypotheses It is not sunny this afternoon and it is colder than yesterday. We will go swimming only if it is sunny. If we do not go swimming, then we will take a canoe trip and If we take a canoe trip, then we will be home by sunset lead to the conclusion We will be home by sunset. (10 markah / marks) BSB/AA Page 10 of 13

11 Soalan / Question 3 Selesaikan yang berikut: / Solve the following: a) Diberi: / Given: A = 1 1 B = Cari / find A B dan/and A B. (4 markah/marks) b) Diberi: / Given: A = B = Cari:/ Find: A B. (6 markah/marks) Soalan / Question 4 Buktikan proposisi yang berikut dengan menggunakan kaedah Aruhan Matematik untuk semua nombor integer bukan negatif. Prove the following propositions by using Mathematical Induction method for all nonnegative integer. P(n) = n = 3( 5 n+1-1) 4 (10 markah/marks) BSB/AA Page 11 of 13

12 Soalan / Question 5 a) Andaikan R 1 = {(1, 1), (1, 2), (2, 3), (3, 1), (3, 3)} dan R 2 = {(1, 2), (1, 3), (2, 1), (3, 3)}. Cari: Let R 1 = {(1, 1), (1, 2), (2, 3), (3, 1), (3, 3)} and R 2 = {(1, 2), (1, 3), (2, 1), (3, 3)}. Find: i. R 1 R 2 ii. R 1 R 2 iii. R 1 - R 2 iv. R 2 - R 1 (4 markah/marks) b) Senaraikan pasangan teratur di dalam hubungan dari X = {1, 2, 3} kepada Y = {a, b, c, d} di mana persembahan matriknya seperti di bawah. / List the ordered pairs in the relations from X = {1, 2, 3} to Y = {a, b, c, d} whose matrix representation is shown below M = (4 markah/marks) BSB/AA Page 12 of 13

13 c) Dapatkan hubungan R ke atas set A = {1, 2, 3, 4} daripada graf berarah di bawah. / Determine the relation R on a set a = {1, 2, 3, 4} from directed graph below. (2 markah/marks) Soalan / Question 6 Apakah kemungkinan Ahmad, Bakar, Charles dan Daud memenangi hadiah pertama, kedua, ketiga dan keempat di dalam satu pertandingan melukis jika hanya 50 orang memasuki pertandingan tersebut dan What is the probability that Ahmad, Bakar, Charles and Daud win the first, second, third, and fourth prizes, respectively, in a drawing contest if only 50 people enter the contest and a) Tiada siapa boleh memenangi lebih dari satu hadiah? No one can win more than one prize? (5 marks/markah) b) Memenangi lebih dari satu hadiah adalah dibenarkan? Winning more than one prize is allowed. (5 marks/markah) End of Page. BSB/AA Page 13 of 13

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