M131: Discrete Mathematics Tutor Marked Assignment Cut-Off Date: December ___, 2013 Total Marks: 40 Contents Feedback form ……….……………..…………..…………………….…...….. 2 Question 1 ……………………..………………………………………..……… 3 Question 2 ……………………………..………………..……………………… 3 Question 3 ………………………………..………………..…………………… 4 Question 4 ………………..……………………………………..……………… 4 Question 5 ……………………..………………………………………..……… 5 Question 6 ……………………………..………………..……………………… 5 Question 7 ………………………………..………………..…………………… 6 Question 8 ………………………………..………………..…………………… 6 Plagiarism Warning: As per AOU rules and regulations, all students are required to submit their own TMA work and avoid plagiarism. The AOU has implemented sophisticated techniques for plagiarism detection. You must provide all references in case you use and quote another persons work in your TMA. You will be penalized for any act of plagiarism as per the AOUs rules and regulations. Declaration of No Plagiarism by Student (to be signed and submitted by students with TMA work): I hereby declare that this submitted TMA work is a result of my own efforts and I have not plagiarized any other persons work. I have provided all references of information that I have used and quoted in my TMA work. Name of Student: ___________________ Signature: _________________ Date: ___________ M131 TMA Feedback Form [A] Student Component Student Name: ____________________ Student Number: ___________________ Group Number: __________ [B] Tutor Component Tutor Name: QUESTION 1 2 3 4 5 6 7 8 MARK 5 5 5 5 5 5 5 5 SCORE TOTAL Tutor’s Comments: The TMA covers only chapters 1, 2, 4 and 9 and consists of eight questions for a total of 40 marks. Please solve each question in the space provided. You should give the details of your solutions and not just the final results. Q–1: [2+3 Marks] a) Find a proposition using only and the connective ˅ with the following truth table: p q ? F F F F T T T F T T T F Answer: b) Using the truth table, determine whether or not the proposition is a tautology. Answer: Q−2: [5×1 marks] Determine whether each of the following is TRUE or FALSE: a) (1 > 2 or 1 + 2 = 4) if (1 + 1 = 2 and 1 > 2). b) (124 mod 6 = 4) → (5 | 16). c) , domain is the set of integers. d) , domain is the set of integers. e) . Answer: a) b) c) d) e) Q¬−3: [3+2 marks] a) Let A = {a, b, c} and B = {1, 2, 3, 4}. Determine whether each of the following is TRUE or FALSE: i. . ii. . iii. . iv. . v. . vi. . Answer: i. ii. iii. iv. v. vi. b) Show that . Answer: Q¬−4: [3+2 marks] a) Consider the decimal number a = 137. i. Find in set builder notation the set of all positive integers b such that b ≡ a (mod 5). ii. Is the number a prime? Explain. iii. Convert the number a to binary and octal numbers. Answer: b) Find the hexadecimal representation of the expansion . Answer: Q−5: [2+3 marks] a) Find the smallest positive integer a in the encryption function f (x) = (ax + 7) mod 26, 0 ≤ x ≤ 25, such that the function encodes the letter “H” by “C”. Answer: b) Decrypt the message “QHHG KHOS” taking into consideration that you are using the encryption function f (x) = (x + 3) mod 26, 0 ≤ x ≤ 25. Answer: Q¬−6: [2+3 marks] Let R1 = {(x, y): |x - y| ≤ 1} and R2 = {(x, y): 2x + y ≤ 6} be relations on the set A = {1, 2, 3, 4}. a) List the elements of R1 and R2. Answer: b) Find and . Answer: Q¬−7: [3+2 marks] a) Find the transitive closure of R = {(a, a), (b, a), (b, c), (c, a), (c, c), (c, d), (d, a), (d, c)} on the set {a, b, c, d}. Answer: b) Find the smallest equivalence relation on {1, 2, 3} that contains (1, 2) and (2, 3). Answer: Q¬−8: [2+3 marks] a) Let R = {(1, 1), (1, 3), (1, 4), (2, 2), (3, 1), (3, 3), (3, 4), (4, 1), (4, 3), (4, 4), (5, 5), (6, 6)} be an equivalence relation on A = {1, 2, 3, 4, 5, 6}. Find the equivalence classes for the partition of A given by R. Answer: b) Let R be the partial order relation defined on A = {2, 3, 4, 5, 6, 8, 10, 40}, where xRy means x | y. i. Draw the Hasse diagram for R. ii. Find the upper and lower bounds of {4, 8}. Answer:
Posted on: Wed, 20 Nov 2013 23:35:43 +0000
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