DBM2023: DISCRETE MATHEMATICS
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1 SECTION A : 50 MARKS BAHAGIAN A : 50 MARKAH INSTRUCTION: This section consists of THREE (3) structured questions. Answer TW0 (2) questions only. ARAHAN : Bahagian ini mengandungi TIGA (3) soalan berstruktur. Jawab DUA (2) soalan sahaja. QUESTION 1 SOALAN 1 CLO1 C1 a) Determine whether the sentence or mathematical expression is proposition or not a proposition. Tentukan samada ayat atau persamaan matematik adalah pernyataan ataupun bukan. i. Tuesday is the day after Wednesday. [1 mark] Selasa adalah hari selepas Rabu. ii. I love that movie very much! [1 mark] Saya sangat sukakan filem itu! iii. 4n + 5 > 12, if n =2 [1 mark] 4n + 5 > 12, jika n =2 iv. Please open the door for the tourist. [1 mark] Tolong bukakan pintu untuk pelancong. v. Is that your favourite car? [1 mark] Adakah itu kereta kegemaran awak? 2 SULIT
2 CLO1 C2 b) If P, Q and R represent the propositions below, write the following propositions in symbolic form. P: You work hard Q: You will get a good salary. R: You can buy a big house. Jika P, Q dan R adalah pernyataan seperti dibawah, tulis pernyataan berikut dalam bentuk simbol. P: Anda bekerja bersungguh-sungguh. Q: Anda akan mendapat gaji yang lumayan. R: Anda boleh membeli rumah yang besar. i. If you do not work hard, then you will not get a good salary. Jika anda tidak bekerja bersungguh-sungguh, anda tidak akan mendapat gaji yang lumayan. [1 mark] ii. You can buy a big house if and only if you work hard and you will get a good salary. Anda boleh membeli rumah yang besar jika dan hanya jika anda kerja bersungguh-sungguh dan mendapat gaji yang lumayan. [2 marks] iii. If you do not work hard, you cannot buy a big house or not getting a good salary. Jika anda tidak bekerja bersungguh-sungguh, anda tidak dapat membeli rumah yang besar atau tidak mendapat gaji yang lumayan. [2 marks] 3 SULIT
3 c) Construct the truth table for [~p (p q)] q and determine whether it is tautology, contradiction or not both. Bina jadual kebenaran bagi [~p (p q)] q dan tentukan samada ia adalah tautologi, kontradiksi atau bukan kedua-duanya. [6 marks] [6 markah] d) Express the following arguments using suitable symbols. Determine whether the argument is valid or not. If it is valid, identify the rule of inference. Nyatakan hujah berikut dengan menggunakan simbol yang sesuai. Tentukan samada hujah tersebut sah atau tidak. Jika sah, kenalpasti hukum ketaakulan. i. Aishah is a student majoring in mathematics. Therefore Aishah is a student majoring in mathematics or computer science. Aishah ialah pelajar major matematik. Oleh itu, Aishah ialah pelajar major matematik atau sains komputer. [3 marks] [3 markah] ii. If it is raining, then the pool will be closed. It is raining. Therefore the pool will be closed. Jika ia hujan, maka kolam akan ditutup. Ia hujan. Maka kolam akan ditutup. [3 marks] [3 markah] iii. If Sara is a doctor, then Sara is married. Sara is a doctor. Therefore, Sara is not married. Jika Sara seorang doktor, maka Sara telah berkahwin. Sara ialah seorang doktor. Maka, Sara tidak berkahwin. [3 marks] [3 markah] 4 SULIT
4 QUESTION 2 SOALAN 2 C2 a) Given set A = {1,2,4}, set B = {3,4,5,6}, set C = {0,1,2,4,6,7,8,9,10} and universal set = {A B C}. Diberi set = {1,2,4}, set B = {3,4,5,6}, set C = {0,1,2,4,6,7,8,9,10} dan set semesta U = {A B C}. i. Draw the Venn diagram that shows all the elements of the universal set, set A, set B and set C. Lukiskan gambarajah Venn yang menunjukkan semua elemen di dalam set semesta, set A, set B dan set C. [4 marks] [4 markah] ii. List all the elements of (A B) C. Senaraikan semua elemen (A B) C. [2 marks] iii. How many elements are there in (B A) (A B)? Berapa bilangan elemen yang ada dalam (B A) (A B)? [4 marks] [4 markah] b) Given A = {1,2,3,4}, B = {1,4,6,8,9} where element a in A is related to element b in B, if and only if, b = a 2. Diberi A = {1,2,3,4}, B = {1,4,6,8,9} di mana elemen a dalam A adalah berhubung dengan elemen b dalam B, jika dan hanya jika, b = a 2. i. Draw the directed graph for the relation. [3 marks] Lukis graf berarah untuk hubungan itu. [3 markah] 5 SULIT
5 ii. Determine whether the relation is reflexive or not. Explain your answer. Tentukan sama ada hubungan itu refleksif atau tidak. Terangkan jawapan anda. [2 marks] iii. Determine whether the relation is symmetric or not. Explain your answer. Tentukan sama ada hubungan itu simetri atau tidak. Terangkan jawapan anda. [3 marks] [3 markah] iv. Determine whether the relation is transitive or not. Tentukan sama ada hubungan itu transitif atau tidak.. [1 marks] c) Given f(x) = x and g(x) = x 3 Diberi f(x) = x dan g(x) = x 3 i. Solve the value of f(g(2)) [2 marks] Selesaikan nilai f(g(2)) ii. Determine the values of x if g(f(x)) = 7 [2 marks] Tentukan nilai x jika g(f(x)) = 7 iii. Find g 1 (5) [2 marks] Cari g 1 (5) 6 SULIT
6 QUESTION 3 SOALAN 3 CLO2 C1 a) Draw a complete bipartite graph of Lukis graf bipartite lengkap bagi i. K 2,3 [2 marks] ii. K 3,4 [2 marks] CLO2 C2 b) Draw a graph for the following set of vertices, V and edges, E. Then determine the degree and parity for of each vertex. Lukiskan graf bagi set bucu, V dan sisi, E berikut. Kemudian tentukan darjah dan pariti bagi setiap bucu. V { A, B, C, D}, E {{A, B}, {A, D}, {B, C}, {B, D}, {C, D}} [6 marks] [6 markah] CLO2 c) Based on the graph in Figure 3(c) below; Berdasarkan graf di dalam Rajah 3(c) di bawah; Figure 3(c) / Rajah 3(c) 7 SULIT
7 i. Determine the parity (even or odd) for each vertex. [2 marks] Tentukan pariti (genap atau ganjil) bagi setiap bucu. ii. Identify the leaf vertex. [1 mark] Kenalpasti bucu daun. iii. Identify the pendant edge. [2 marks] Kenal pasti sisi loket. iv. Identify the distance from a to f. [2 marks] Kenalpasti jarak dari a ke f. v. Identify the size of the graph. [2 marks] Kenalpasti saiz graf. vi. Find two (2) simple paths from a to c. [2 marks] Dapatkan dua (2) laluan ringkas dari a ke c. vii. Find a trail from d to e. [2 marks] Dapatkan jejak dari d ke e. viii. Find two (2) simple cycles labeled as C 1 and C 2. [2 marks] Cari dua (2) kitaran mudah, dilabel sebagai C 1 dan C 2. 8 SULIT
8 SECTION B : 50 MARKS BAHAGIAN B : 50 MARKAH INSTRUCTION: This section consists of THREE (3) structured questions. Answer TW0 (2) questions only. ARAHAN : Bahagian ini mengandungi TIGA (3) soalan berstruktur. Jawab DUA (2) soalan sahaja. QUESTION 4 SOALAN 4 CLO 2 C1 a) Answer the following questions based on the tree shown in Figure 4(a). Jawab soalan berikut berdasarkan pokok yang ditunjuk di Rajah 4(a). a b c d e f g h i j k l m Figure 4(a) / Rajah 4(a) i. Find all the internal vertices. [1 mark] Cari semua bucu dalaman. ii. Find the ancestors of k. [1 mark] Cari nenek moyang k. 9 SULIT
9 iii. List all the vertices of level 2. [1 mark] Senarai bucu pada tahap 2. iv. What is the height of the tree? [1 mark] Apakah ketinggian pokok? CLO 2 C2 b) Find the minimum spanning tree of the weighted graph shown in Figure 4(b) by using Prim s algorithm. Cari pokok merentang minimum untuk graf pemberat yang diberi dalam Rajah 4(b) dengan menggunakan algoritma Prim. [6 marks] [6 markah] Figure 4(b) / Rajah 4(b) CLO 2 c) Build a binary search tree for the words Hockey, Football, Golf, Squash, Tennis, Futsal, Baseball, Volleyball, Takraw, Basketball and Pool using alphabetical order. Bina pokok carian dedua untuk perkataan Hockey, Football, Golf, Squash, Tennis, Futsal, Baseball, Volleyball, Takraw, Basketball dan Pool dengan menggunakan urutan abjad. [6 marks] [6 markah] 10 SULIT
10 CLO 2 d) Find the order of vertices of the tree shown in Figure 4(d) by using preorder, postorder and inorder traversal. Cari susunan bucu bagi pokok dalam Rajah 4(d) dengan menggunakan preorder, postorder dan inorder traversal. [9 marks] [9 markah] Figure 4(d) / Rajah 4(d) 11 SULIT
11 QUESTION 5 SOALAN 5 C2 a) Suppose that f is recursively defined by f(0) = 3, f(1) = 1 and f(n + 1) = f(n) 2f(n 1) + 5. Find: Fungsi f ditakrifkan dengan f(0) = 3, f(1) = 1 dan f(n + 1) = f(n) 2f(n 1) + 5. Dapatkan: i. f(2) [2 marks] ii. f(3) [2 marks] iii. f(4) [2 marks] iv. f(5) [2 marks] v. f(6) [2 marks] b) i. Prove using mathematical induction that for all n 1, n 3 = [ n(n+1) ] 2. 2 Buktikan n 3 = [ n(n+1) ] 2, n 1, dengan menggunakan 2 kaedah induksi matematik. [7 marks] [7 markah] 12 SULIT
12 ii. Give a recursive definition for the following if n = 1,2,3 Berikan definisi rekursif bagi persamaan berikut jika n = 1,2,3 a. K n = 6n [4 marks] [4 markah] b. K n = 2n + 1 [4 marks] [4 markah] 13 SULIT
13 QUESTION 6 SOALAN 6 CLO1 C2 a) Give the possible combination for the given situations. Berikan kombinasi yang mungkin bagi situasi berikut. i. Ayuni has 2 types of skirt, 2 blouses and 3 pairs of shoes. How many different outfit combinations of one skirt, one top and one pair of shoes Ayuni have? Ayuni mempunyai 2 jenis skirt, 2 blaus dan 3 pasang kasut. Berapakah kombinasi pakaian yang berlainan yang dipunyai oleh Ayuni dengan satu skirt, satu blaus dan sepasang kasut? [3 marks] [3 markah] ii. Ray Resturant has 3 types of fishes, 2 types of side dishes and 3 types of drinks. How many different meal combinations of 1 fish, 1 side dish and 1 drink that the restaurant offer? Restoran Ray mempunyai 3 jenis ikan, 2 jenis hidangan sampingan dan 3 jenis minuman. Berapakah kombinasi hidangan berbeza yang mengandungi 1 ikan, 1 hidangan sampingan dan 1 minuman yang ditawarkan di restoran tersebut? [3 marks] [3 markah] iii. California Steak House offers a 3 course meal that consists of an appetizer (onion rings, fries and popcorn), a main dish (steak, chicken, pasta) and a dessert (sundae, pie, cheese cake, chocolate cake). How many different combination of one appetizer, one main course and one dessert the restaurant offers? California Steak House menawarkan 3 hidangan yang terdiri daripada pembuka selera (gegelang bawang, kentang goreng dan bertih jagung), hidangan utama (stik, ayam dan pasta) dan pencuci mulut (sundae, pai, kek keju dan kek coklat). Berapakah bilangan gabungan hidangan dengan satu pembuka selera, satu hidangan utama dan juga satu pencuci mulut yang ditawarkan oleh restoran tersebut? [4 marks] [4 markah] 14 SULIT
14 b) Find the total number of outcome for the situation below: Hitung jumlah bilangan kesudahan bagi situasi di bawah: i. If a three digit number is formed from the digits 1,2,3,4,5,6 and 7 without repetitions, calculate how many of these three digit numbers will have a number value between 100 and 500. Sekiranya tiga digit dibentuk daripada digit-digit 1,2,3,4,5,6 dan 7 tanpa pengulangan, hitungkan berapa cara daripada tiga digit yang dibentuk itu mempunyai nilai di antara 100 dan 500. [5 marks] [5 markah] ii. How many ways can 4 girls and 5 boys be arranged in a row so that all the 4 girls are together? Berapakah bilangan cara supaya 4 budak perempuan dan 5 budak lelaki boleh disusun dalam satu baris agar keempat-empat budak perempuan boleh berada bersama-sama? [5 marks] [5 markah] iii. A door can be opened only with a security code that consists of five buttons: 1, 2, 3, 4, 5. A code consists of pressing any one button, or any two, or any three, or any four, or all five. How many possible codes are there? Sebuah pintu hanya boleh dibuka dengan kod sekuriti yang mengandungi 5 butang iaitu 1,2,3,4 dan 5. Kod itu boleh dibuka dengan hanya menekan mana-mana satu butang, atau mana-mana dua butang,atau mana-mana tiga,mana-mana empat atau kesemua 5 butang. Berapakah bilangan kemungkinan kod tersebut? [5 marks] [5 markah] SOALAN TAMAT 15 SULIT
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