M131: Discrete Mathematics Tutor Marked Assignment Cut-Off Date: May ___, 2014 Q–1: [2+3 marks] a) Find a proposition with three variables p, q, and r that is never true using only . b) Show that the statement is a tautology by using truth tables and without using truth tables. Q−2: [5×1 marks] Write each of the following statements in the form “If …, then …”: a) x is even only if y is odd. b) To get a good grade it is necessary that you study. c) Studying is sufficient for passing. d) It is not Thursday or it is not cold. e) You cannot not either register or check out library books. Q¬−3: [3+2 marks] a) Determine whether each of the following is TRUE or FALSE: i. , domain is the set of real numbers. ii. , domain is the set integers. iii. For all z (z is prime and 2 z). iv. If , then A = B. v. If , then B = A. vi. A (B C) = (A B) C. b) Determine whether or not . Justify your answer. Q¬−4: [2+2+1 marks] a) List all positive integers less than 30 that are relatively primes to 20. b) Determine whether or not the statement “for all integers a, b, c, if a | bc, then a | b or a | c” is false. Justify your answer. c) Suppose that the lcm of two numbers is 400 and their gcd is 10. If one of the numbers is 50, find the other number. Q−5: [5 marks] Decrypt the message “YZUV GZ TUUT” using the encryption function f (x) = (x + 6) mod 26, 0 ≤ x ≤ 25. Q¬−6: [1+2+2 marks] Let R = {(x, y): x < y +1} be a binary relation on the set A = {1, 2, 3, 4}. a) List the elements of R. b) Find the symmetric and transitive closures of R. c) Find . Q¬−7: [5×1 marks] Suppose that R and S are relations on the set A = {a, b, c, d}, where R = {(a, b), (a, d), (b, c), (c, c), (d. a)} and S = {(a, c), (b, d), (d, a)}. a) Construct R2. b) Construct R3. c) Construct S3. d) Construct R○S. e) Construct S○R. Q¬−8: [5 marks] Suppose that R and S are equivalence relations on a set A. Show that the relation R ∩ S is also an equivalence relation on A.
Posted on: Fri, 14 Mar 2014 12:10:36 +0000
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