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Institute of Business Administration Faculty of computer Science Department of Mathematical Sciences First Term Exam Logic and Discrete Structures March 08, 2012 Student Name:_______________________________________________ Note:- Please solve ALL FIVE questions which are equally weighted. Q1 Decide whether each of the following is tautology, contradiction or a contingency. a) [p (q q)] p b) (pq) q c) p qq Q2 Determine whether these arguments are valid. (You are not supposed to use truth tables here.) To prove that an argument is valid you must use 19 rules of logic. For an invalid argument you are required to provide a counter example. a) b) p q p q p r c) p q r q p ________________________ p (q r) _____________ ________________ q p r Q3 Let x be an integer. Prove that if 5x2 – 3 is odd, then x is even. Q4 Determine the truth value of each of these statements if the domain of the variable x consists of all real numbers. a) x(x2 = 2) b) x(x2 = –1) c) x(x2 + 2 ≥ 1) d) x(x2 ≠ x) Q5 Let A = {1, 2, 3, 4}. Consider the following binary relations on A R1 = { (1,2), (1,3), (3,1), (1,1), (3,3), (1,4), (4,2), (3,4) } R2 = { (2,2), (1,3), (1,1), (3,3), (3,2), (1,4), (4,2), (4,4) } R3 = { (3,3), (4,4), (1,3), (1,4), (4,2), (1, 2), (3,4), (1, 1), (2, 2) } a) Which of these relations are reflexive? Please explain. b) Which of these relations are transitive? Please explain. c) Write down the inverse relation of R1. d) Write down the composition R2 o R3 of relations R3 and R2. 1 2 3 4