CSCI 3310 - Homework Set 1 Due: September 11, 2018 at the beginning of class ANSWERS Please write your name and student ID number clearly at the top of your homework. If you have multiple pages, please make sure they are secured together. Problem 1 Indicate whether each statement below is a proposition. If the statement is a proposition, express the negation of the proposition in English or mathematical symbols. a. 0 < x < 10. No b. π is approximately 22/7. Yes; π is not approximately 22/7 c. Pay attention in class. No d. All years divisible by 4 are leap years. Yes; Not all years divisible by 4 are leap years. Problem 2 he propositional variables, p, q, and s have the following truth assignments: p =, q = F, r = F. Give the truth value for the following compound propositions: a. p q b. q r F c. p r F d. r Problem 3 he values for the propositional variables p, q, r, s, are determined as follows: p = q = F r = F s = Determine the truth values for the following propositions: a. (p q) s b. (r q) p c. (s q) (p q)
d. (r p) s e. p (q s) Problem 4 Give a truth table for the following propositions: a. (p q) (q p) p q p q q p (p q) (q p) F F F F F F F F b. p ( p q) p q p p q p ( p q) F F F F F F F F F F c. (p r) (r q) p q r r p r (r q) (p r) (r q) F F F F F F F F F F F F F F F F F F F F F F F F F Problem 5 Let p and q be the propositions p : I bought a lottery ticket this week. q : I won the million dollar jackpot. Express each of these propositions as an English sentence. a. p q I bought a lottery ticket this week or I didn t win the million dollar jackpot b. p q If I bought a lottery ticket this week, then I won the million dollar jackpot
c. p q I bought a lottery ticket this week if and only if I won the million dollar jackpot d. p q If I didn t buy a lottery ticket this week then I will not win the million dollar jackpot e. p q I didn t buy a lottery ticket this week and I didn t win the million dollar jackpot f. p (p q) I didn t buy a lottery ticket this week or I bought a lottery ticket this week and won the million dollar jackpot. Problem 6 Let p, q, and r be the propositions p : You get an A on the final exam. q : You do every exercise in this book. r : You get an A in this class. Write these propositions using p, q, and r and logical connectives (including negations). a. You get an A in this class, but you do not do every exercise in this book. r q b. You get an A on the final, you do every exercise in this book, and you get an A in this class. p q r c. o get an A in this class, it is necessary for you to get an A on the final. p r d. You get an A on the final, but you don t do every exercise in this book; nevertheless, you get an A in this class. p q r e. Getting an A on the final and doing every exercise in this book is sufficient for getting an A in this class. (p q) r Problem 7 Find out for each of the following propositions whether it is a tautology, a contradiction, or neither (a contingency). Prove your answer (use a truth table).
a. [(p q) (q r)] (p r) p q r p q q r (p q) (q r) p r [(p q) (q r)] (p r) F F F F F F F F F F F F F F F F F F F F F F So it is a tautology. b. (p q r) [(q r) (p q)] p q r p q r p q q r (q r) (p q) (p q r) [(q r) (p q)] F F F F F F F F F F F F F F F F F F F F F F F F F So it is a contingency. Problem 8 For each table, give a logical expression whose truth table is the same as the one given below (note that there may be several correct answers): a. p q? F F F F F F F his could be p q or ( p q) or
b. p q? F F F F F his could be p q or ( p q) or Problem 9 Consider the following sentence: here is a student in this class who has taken some course in every department in the College of Engineering and Computer Science. a. Express the sentence using quantifiers (and logical propositions). You should use the variables: s student, c course, d department, and these should be the only variables. Give the domain of each variable. b. Find the logical negation of the quantifier expression that you obtained in (a). c. ranslate the negation you obtained in (b) into an English sentence. a. Define the following variables and domains Variable Domain s student this class c course all courses d department departments in the College We define the propositional function: P(s, c, d) = s took course c in department d. hen the sentence can be written as s d c P(s, c, d). b. he negation is s d c P(s, c, d). c. In English: Every student in this class has not taken any course in some department in the College.