Mathematics for Computer Science MATH 1105C Page 1 of 5

Transcription

1 x Mathematics for Computer Science MATH 1105C Page 1 of 5

2 ANSWER ALL QUESTIONS Question 1: (25 Marks) Construct the truth table of proposition (p q) ( p q). Determine if this proposition is a tautology. (b) Construct a truth table for (t u) v. (c) Show that (p q) p q. (d) Use truth table to show s r r s. (e) Use proof by cases to show that if w is a positive integer then w 3 + w is even. (f) Use induction to show that (t u) is always a factor of S n where S n = t n u n, n 1. Mathematics for Computer Science MATH 1105C Page 2 of 5

3 Question 2: (25 Marks) Over athletes competed in 2010 International Games in America. The International Game Committee wanted to ensure that the competition administered more than 5000 drug tests to athletes. All the medal winners were tested, as well as other randomly selected competitors. Suppose that 2% of athletes had actually taken banned drug. No drug test is perfect. Sometimes the test says that an athlete took drugs, but the athlete actually did not. We call this a false positive test result. Other times, the drug tests says that an athlete is Clean but the athlete actually took drugs. This is called a false negative result. Suppose that the testing procedure used at the International Game has a false positive rate of 1% and a false negative of 0.5%. What is the probability that an athlete who tests positive actually took drugs? (b) Show that A and B are independent if and only if P (A B) = P (A) P (B). (3 marks) (c) A box contains 7 bananas and 5 apples. The fruits are taken out of the box, one at a time and in a random order. What is the probability that the box is empty after the last banana is taken from the box? (d) The probability that after a visit to ABCD car dealer results in neither buying a second-hand car nor a European car is 55%. Among those coming to the dealer, 25% buy a second-hand car and 30% buy a European car. What is the probability that a visit leads to buying a second-hand European car? (e) Given that a 0 = 2, a n+1 = 3a n for n 0, use generating function to find a n in terms of n. (f) Show that the set Z of integers together with the relation of inequality is a poset. (3 marks) Mathematics for Computer Science MATH 1105C Page 3 of 5

4 Question 3: (25 Marks) (Egypt Die Hard Problem) Wandering in the lands of Egypt, you get the opportunity to visit a pyramid. Unknowingly, you press a hidden switch which lands you in a trapped room. In the room, there is only one locked door to exit. Apart from that, there is a fountain from which a dark liquid is coming out. Next to it there are two empty jugs, a 7 litres and a 5 litres. At a distance of 5 steps away from the fountain, there is an uneven tile which is 5cms above the floor level. On the wall instructions of how to exit the room are etched. They are shown below: Fill on of the jugs with exactly 6 litres of the dark liquid and place it on the uneven tile in order to open the locked door. You must be precise as one ounce more or less will result in the crumbling of the roof followed by filling the space with sand. If you do not succeed to complete the task in 10 minutes, the roof will automatically collapse and fill the space with sand. (i) (ii) (iii) Model jug filling scenarios with a state machine for the Egypt Die Hard problem. Clearly, indicate the start state and the kinds of transitions. (12 marks) Provide difference between the 99-Counter state machine and the Egypt Die Hard machine. State whether the Egypt Die Hard machine is deterministic or non deterministic and explain why it is so. (2 marks) (b) Show that any Y = {y 1, y 2,..., y x } of x integers has a subset such that the sum of the integers in the subset is divisible by x. (c) Use the product rule to show that P (n, r) = n! (n r)!. Hence, find all possible 4-permutations on the set {a, b, c, d, e, f, g, h, i, j}. Mathematics for Computer Science MATH 1105C Page 4 of 5

5 Question 4: (25 Marks) Consider the following incidence matrix: e 1 e 2 e 3 e 4 e 5 e 6 v v v v v v v (i) Draw the graph of the incidence matrix. (ii) List the isolated vertices. (iii) List the loops. (iv) List the parallel edges. (v) List the vertices adjacent to v 3. (vi) Find all edges incident on v 4. (b) Draw K 3, K 3,4 and K 2,5. (c) (i) Show that the composition of two injective functions is also injective. (6 marks) (ii) (iii) Show that the composition of two surjective functions is also surjective. Show that the composition of two bijective functions is also bijective. ***END OF QUESTION PAPER*** Mathematics for Computer Science MATH 1105C Page 5 of 5