(c) Give a proof of or a counterexample to the following statement: (3n 2)= n(3n 1) 2
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1 Question 1 (a) Suppose A is the set of distinct letters in the word elephant, B is the set of distinct letters in the word sycophant, C is the set of distinct letters in the word fantastic, and D is the set of distinct letters in the word student. The universe U is the set of 6 lower-case letters of the English alphabet. Find (i) A B (ii) A C (iii) A (C D) (iv) (A B C D) (b) Find two finite sets A and B such that A B and A B. (c) Give a proof of or a counterexample to the following statement: A (B C)=(A B) (A C) Question Prove by mathematical induction that, for all n 1, [6 marks] Question (3n )= n(3n 1) [6 marks] Given the function f :N R, f (x)=x 4 + x prove from definition that f (x) Θ(x 4 ). Question 4 (a) Given the proposition p :John is tall and John is thin write down its negation p. (Hint: use De Morgan s laws). (b) Given a proposition p q we know that neither its converse (q p), nor its inverse ( p q) is a consequece of it. Using the truth table method prove that p q and its contrapositive are logically equivalent. (c) The contrapositive proof technique uses the above identity: if we want to prove p q, it is exactly the same thing to prove its contrapositive. Use the contrapositive proof technique to prove that if the square of an integer m is even, then the integer m must be even.
2 Question 5 (a) Rewrite the following formal statement in a variety (at least two) of equivalent but more informal ways (plain English). Do not use the symbols and. x R, x 0 (b) Rewrite the following statement formally. Use quantifiers, variables and predicates. There is a barber who shaves all men in town who do not shave themselves. (c) A universal conditional statement is written formally as x D, if P(x) then Q(x) A universal conditional statement is not logically equivalent to its converse. (i) Write the preceding statement in symbols. (ii) Give an example. Question 6 Construct a table showing the interchanges that occur at each step when selection sort is applied to the following list: 5, 3, 4, 6, Question 7 is on page 4 3
3 Question 7 Consider the following graph: v 1 e 7 e 1 v 6 e e 3 v 4 e 5 v e 4 v 3 v 5 e 6 Figure 1: Your first graph (a) Write the edge set, the vertex set, and give a table showing the edge endpoints functionσ. (b) Find all edges that are incident on v 1, all vertices that are adjacent to v 3, all edges that are adjacent to e 4, all loops, all parallel edges, all vertices that are adjacent to themselves, and all isolated vertices. (c) Show that the graph bellow does (not) have an Euler circuit. e 1 v v 3 e 5 e e 4 v 1 e 6 e 3 v 4 e 7 Figure : Your second graph Question 8 is on page 5 4
4 Question 8 (a) Draw a binary tree to represent the following mathematical expression: (a+b) (c+d) (b) Write the prefix (preorder traversal) and postfix (postorder traversal) forms of the expression (of the binary tree). (c) Use Kruskal s algorithm to find a minimal spanning tree for the following graph, where the numbers represent the weight of the corresponding edges. What is the total weight of the minimal spanning tree? a 9 e b 1 7 c 4 10 d 3 5 f 8 g 6 Figure 3: Another graph Question 9 Given the number X= 13 x, where x is the smallest possible base for it, convert X in bases, 8, 10 and hexa. Question 10 LetSbe a partition of the set X= { 1,, 3, 4, 5, 6 } consisting of three subsets, each of them containing at most three elements of the initial set. Define x R y to mean that for some set S S, both x and y belong to S. (a) Give R explicitly by its elements. (b) Prove that R is an equivalence relation. (c) Draw the digraph of the relation R. Question 11 is on page 6 5
5 Question 11 (a) Draw a gate implementation for a One-Bit Equality Circuit: the output of this circuit is 1 if and only if both inputs are 0 or both inputs are 1. (b) Find the canonical form for f= xy+z. (c) Explicitly define the canonical form for f= xy+z by means of a truth table. Question 1 [7 marks] Let us assume that a and b are given constants and that the two initial values s 0 and s 1 are known for the recurrence relation s n = as n 1 + bs n (a) Find the general solution for the case where b=0. (b) Find the general solution for the case where a=0. (c) Give examples for both cases. Please remember This examination question paper MUST BE HANDED IN. Failure to do so may result in the cancellation of all marks for this examination. Writing your name and number on the front will help us confirm that your paper has been returned. 6
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