# MSS 318 Discrete Mathematics [Matematik Diskret]

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1 UNIVERSITI SAINS MALAYSIA Second Semester Examination 2009/2010 Academic Session April/May 2010 MSS 318 Discrete Mathematics [Matematik Diskret] Duration : 3 hours [Masa : 3 jam] Please check that this examination paper consists of SEVEN pages of printed material before you begin the examination. [Sila pastikan bahawa kertas peperiksaan ini mengandungi TUJUH muka surat yang bercetak sebelum anda memulakan peperiksaan ini.] Instructions : Answer all ten [10] questions. [Arahan : Jawab semua sepuluh [10] soalan.] In the event of any discrepancies, the English version shall be used. [Sekiranya terdapat sebarang percanggahan pada soalan peperiksaan, versi Bahasa Inggeris hendaklah diguna pakai]. 2/-

2 2 1. Defining a word as any string of seven letters of the English alphabet {a,b,, z}, with repeated letters allowed, find the number of words that (i) begin with a vowel and end with a vowel. (ii) begin with a vowel or end with a vowel. (iii) have no vowels. (iv) have exactly one vowel. (v) begin with a or b or end with a or b. 2. Find the number of subsets of S that contain (i) no odd numbers. (ii) exactly five elements, all of them even. (iii) exactly three elements, all of them even. (iv) exactly five elements, including 3 or 4 but not both. (v) exactly five elements, the sum of which is even. 3. A club with 20 women and 17 men needs to form a committee of size six. How many committees are possible if the committee (i) must have three women and three men? (ii) must consist of all women or all men? (iii) must have at least two men? (iv) must have at least one woman? (v) must have at least one man and at least one woman? 4. For the graphs mentioned in the following statements, either give an example or prove that there is no such graph: (i) A simple graph with 6 vertices and 16 edges. (ii) A simple graph with 8 vertices, whose degrees are (iii) A graph with 6 vertices that has an Euler circuit but no Hamilton circuit. (iv) A connected simple planar graph with 5 regions and 8 vertices, each of degree 3. (v) A simple digraph with indegrees: and outdegrees: /-

3 3 1. Dengan menakrifkan "perkataan" sebagai sebarang rantai tujuh huruf dari set {a, b,..., z}, dengan ulangan dibenarkan, cari bilangan perkataan yang (i) bermula dengan vokal dan berakhir dengan vokal. (ii) bermula dengan vokal atau berakhir dengan vokal. (iii) tidak mempunyai vokal. (iv) memiliki tepat satu vokal. (v) bermula dengan a atau b, atau berakhir dengan a atau b. 2. Tentukan bilangan subset dari S yang mengandungi (i) tiada nombor ganjil. (ii) tepat lima unsur, kesemuanya genap. (iii) tepat tiga unsur, kesemuanya genap. (iv) tepat lima unsur, termasuk 3 atau 4 tetapi tidak kedua-duanya. (v) tepat lima unsur, hasil tambah mereka genap. 3. Sebuah kelab dengan 20 perempuan dan 17 lelaki perlu membentuk sebuah jawatankuasa bersaiz enam. Berapakah bilangan jawatankuasa yang mungkin dibentuk sekiranya jawatankuasa itu (i) mesti mempunyai tiga orang perempuan dan tiga orang lelaki? (ii) harus terdiri daripada semua wanita atau semua lelaki? (iii) harus mempunyai sekurang-kurangnya dua orang lelaki? (iv) harus mempunyai sekurang-kurangnya seorang perempuan? (v) harus mempunyai sekurang-kurangnya seorang lelaki dan sekurang-kurangnya seorang perempuan? 4. Untuk setiap graf yang dinyatakan dalam pernyataan berikut, berikan satu contoh graf atau buktikan ketidakwujudannya: (i) Sebuah graf mudah (simple graph) dengan 6 verteks dan 16 tepi. (ii) Sebuah graf mudah (simple graph) dengan 8 verteks, yang berperingkat 0,1,2,3,4,5,6,7. (iii) Sebuah graf dengan 6 verteks yang mempunyai suatu litar Euler tetapi tiada litar Hamilton. (iv) Sebuah graf planar mudah (simple planar graph) disambungkan dengan 5 kawasan dan 8 verteks, setiapnya berperingkat 3. (v) Sebuah digraf mudah (simple digraph) dengan indegree: 0,1,2,2,3,4 dan outdegree: 1,1,2,2,3,4. 4/-

4 4 5. (a) Form a recurrence relation for each of the following sequences and give initial conditions, assuming that the sequences begin with a 1 : (i) (ii) a n the number of bit strings of length n that do not have two consecutive 0s. If G(x) is the generating function for a 0 a 1 a 2 a 3, describe in terms of G(x) the generating function for (i) a a a a 3 (ii) a 0 0 a 1 0 a 2 0 a 3 0 a 4. [20 marks] 6. (a) Determine whether the given pair of graphs in (i) (G 1,G 2 ) (ii) (H 1,H 2 ) is isomorphic. Exhibit an isomorphism or provide an argument for nonisomorphism. If a connected planar simple graph has e edges and v vertices with v 3 and no circuits of length three, then prove that e 2v 4. [20 marks] 7. (a) How many different strings can be made from the letters in ORONO, using some or all of the letters? Eighteen guests have to be seated, half on each side of two sides of a long table. Four particular guests desire to sit on one particular side and three others on the other side. Determine the number of ways in which the sitting arrangements can be made. 5/-

5 5 5. (a) Bentukkan suatu hubungan pengulangan untuk setiap urutan berikut dan berikan syarat awal, dengan anggapan bahawa urutan tersebut bermula dengan a 1: (i) (ii) a n bilangan rantai bit (bit string) dengan panjang n yang tidak mempunyai dua 0 yang berturut-turut. Jika G(x) adalah fungsi penjana bagi a 0 a 1 a 2 a 3, nyatakan dalam sebutan G(x) fungsi penjana untuk (i) a a a a 3 (ii) a 0 0 a 1 0 a 2 0 a 3 0 a 4. [20 markah] 6. (a) Tentukan sama ada pasangan graf yang diberikan dalam (i) (G 1,G 2 ) (ii) (H 1,H 2 ) berisomorfik. Tunjukkan suatu isomorfisma atau berikan pejelasan untuk ketiadaan isomorfisma. Jika sebuah graf planar mudah tersambung mempunyai e tepi dan v verteks dengan v 3 dan tiada rantai yang panjangnya tiga, buktikan bahawa e 2v 4. [20 markah] 7. (a) Berapakah bilangan rantai yang berbeza yang dapat dibina daripada huruf-huruf dalam ORONO, menggunakan beberapa atau kesemua huruf? Lapan belas tetamu harus duduk, setengah di setiap sisi dari dua sisi sebuah meja panjang. Empat tetamu tertentu ingin duduk pada satu sisi tertentu dan tiga orang lain di sisi lain. Tentukan jumlah cara di mana tatacara duduk ini boleh dibuat. 6/-

6 6 8. (a) How many solutions are there to the equation x1 x2 x3 x4 x 5 21, where xi i, 1,2,3,4,5, is a non-negative integer such that i x1 10 ii 0 x 1 10? If n is a positive integer, show that 2n 2n 1 2n 2 n 1 n 2 n (a) Solve the recurrence relation by using the characteristic equation: a 4a 3a 2 n 3, a 1, a 4. n n n 1 n Using generating functions solve a n 3a n 1 2 n 5, a Let G be a simple graph with n vertices. Show that G is a tree if and only if G is connected and has n-1 edges. 7/-

7 7 8. (a) Berapakah bilangan penyelesaian yang wujud bagi persamaan x x x x x di mana xi, 1,2,3,4,5, adalah integer tak-negatif i x 10 ii 0 x 10?. supaya 1 1 i Jika n adalah suatu integer positif, tunjukkan bahawa 2n 2n 1 2n 2 n 1 n 2 n (a) Selesaikan hubungan ulangan berikut dengan menggunakan persamaan cirian: a 4a 3a 2 n 3, a 1, a 4. n n n 1 n Gunakan fungsi-fungsi penjana untuk menyelesaikan a n 3a n 1 2 n 5, a Katakan G sebuah graf mudah dengan n verteks. Tunjukkan bahawa G merupakan sebuah pohon jika dan hanya jika G adalah terkait dan mempunyai n-1 tepi. - ooo O ooo -

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