CSCI/MATH 2112 Discrete Structures I Assignment 1. Due on Friday, January 18, 11:00 pm (1) Write symbolic expression for each of the statements below; then work out their negations; finally expressing each as complete sentence in English: (a) Roses are red, violets are blue. (b) The bus is late or my watch is slow. (c) If a number is prime then it is odd or it is 2. (d) If a number x is a prime, then (root ) x is not a rational number.

(2) On a vacation in the Crazoes islands (recall that all inhabitants of C. are either knights or knaves), you run into three inhabitants, A, B, C. (a) Using a to represent the proposition: A is a knight (and similarly b; c) and the basic logic operations, to write the following in symbolic notation. (i) A is lying and B or C is truthful. (ii) A and B are lying or A and C are truthful. (iii) At least two people are telling the truth. (iv) Exactly two are telling the truth. (b) Solve the following logic puzzles (as far as possible) by using truth tables. Who, if anyone, tells the truth in each case? Explain your answers. (i) A: \B or C is lying." B: \C is lying". C: \A and I are both truthful." (ii) A: \I am lying if B is" B: \A is lying if I am" (iii) A: \If B is lying, then so is C" B: \If C is lying, then so is A" C: \If A is lying, then so is B"

(3) (a) Show ~((p or ~ q) ^ r) = (~ p ^ q)or ~r Identify each Boolean algebra rule (see Boolean Identities posted) used. (b) You are given that for a particular assignment of truth values, the Boolean expression (p^q)orr ) ) (r or s) is false. Without using a truth table, and the truth values of p; q; rs. Clearly explain your answer.

(4) Using only the operators ) ~ and =>, Find the equivalent Boolean functions of: (a) p or q (b) p ^ q (c) p =(3line equal) q (this is equivalent to : p <=> q)

(5) For n E(epsilon) N ,which of the following : (i) 3 | n (ii) 9 | n (iii) 3 | n² (iv) 12 | n (v) n even ^ 3 | n ( Note: Read a | b as "a divides b") (a) is (are) necessary for n to be divisible by 6? Briefly explain each answers. (b) is (are) suffcient for 6 | n? Brie y explain each answers.