UNIVERSITY EXAMINATION 2018/2019 YEAR I SEMESTER I EXAMINATION FOR THE DEGREE OF BACHELOR OF SCIENCE IN MATHEMATICS AND COMPUTER SCIENCE, ACTURIAL SCIENCE, STATISTICS AND COMPUTER SCIENCE SPM 2100: DISCRETE MATHEMATICS Date: Thursday, 13th December 2018 Time: 11.00am β 1.00pm INSTRUCTIONS Answer question one (compulsory) and any other two questions Question One (30 Marks) a) Define the following term (i) Tautology (2 marks) (ii) Contingency (1 mark) (iii) (2 marks) A Set b) Let A = {4, 2, 6, 1} and B = {a, ,5,4}. Evaluate: (i) B-A (2 marks) (ii) π΄ β© π΅ (1 marks) (iii) |π΄ βͺ π΅| (2 marks) c) Show that if x is odd then x 2 is odd (3 marks) d) Determine the truth value of the following sentences (i) If 1 + 1 = 2, then some birds can fly (ii) Nairobi is the only city in kenya and 2 + 3 = 6 (iii) (2 marks) (2 marks) 1 + 1 = 31 if and only if monkeys can fly (2 marks) e) Determine the validity of the following argument. (4 marks) (π β§ π) β π π π f) Prove β3 is irrational using contradiction (5 marks) g) Let f: β€ β β€ with f (x) = π₯ 2 . Find the domain and the corresponding range of f (2 marks) Question Two (20 Marks) a) Proof by mathematics induction that Page 1 of 3 1 + 2+ 3 + β―+ π = π(π+1) (7 marks) 2 b) Use truth tables to test if the following is a tautology, a contradiction or a contingency. (7marks) c) Prove using contrapositive "if 3n + 2 is odd, then n is odd" (4 marks) d) Negate the statement βπ₯πβ, βπ¦πβ|π₯ + π¦ = 0 (2marks) Question Three (20 Marks) a) Let π, β: β β β such that f (x) = 3x + 2 and h(x) = x 2 + 2x + 1. Find (i) ( βππ)(π₯) (3 marks) (ii) (ππβ)(2) (3 marks) (iii) π β1 (π₯) (2 marks) (iv) Is π bijective mapping (3 marks) b) State the necessary and sufficient condition of the following statement βif someone is a mother, she must be a ladyβ. (3 marks) c) Write the inverse, converse and contrapositive of the following statement "if 2 + 3 = 5, then mathematics is easy" (6 marks) Question Four (20 Marks) a) Let A = {1,2,3,4} and B = {a, b, c, d). Find (i) P(B) and |P(B)|,where P(B) is the power set of B (5 marks) (ii) AxB (3 marks) b) In a survey including 60 people, 25 take milk, 26 tea and 26 coffee, 9 like both milk and tea, 11 like milk and coffee, 8 like coffee and tea and 8 like none of the three drinks (i) Draw a vehn diagram representing then information (ii) Find the number of people who like all the three drinks (iii) Find the number of people who like exactly one of the (4 marks) (2 marks) (6 marks) three drinks. Question Five (20 Marks) a) Consider the following propositions p: mathematicians are generous q: Spiders hate algebra write the compound propositions symbolized by Page 2 of 3 (i) ππ¬π (2 marks) (ii) ¬(πβπ) (2 marks) (iii) ¬π β ¬π (2 marks) (iv) ¬π βΊ ¬π (2 marks) b) Prove that (π΄ππ΅)π = π΄π ππ΅ π (4 marks) c) (i) Define the following functions 1. One-to one functions (2 marks) 2. Onto functions (2 marks) (ii) Determine whether the following functions are injective or surjective functions 1. π: β β β when π(π₯) = 3π₯ + 7 for all π₯πβ (2 marks) 2. π: β β β when π(π₯) = π₯ 3 for all π₯πβ (2 marks) Page 3 of 3