Concordia University Faculty of Engineering and Computer Science

COEN231: Introduction to Discrete Mathematics Fall, 2011  Homework 1 Due date: Tuesday September 27, 2011 1. Let p, q and r be the propositions p: you have the flu q: you miss the final examination r: you pass the course Express each of these propositions as an English sentence

• p → q • ¬q ↔ r • q → ¬r • (p → ¬ r) ∨ (q → ¬r)

2. 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 logical connectives

• To get an A in this class, it is necessary for you to get an A on the final • Getting an A on the final and doing every exercise is this book is sufficient for getting an A in

this class • You will get an A in this class if and only if you either do every exercise in this book or you get

an A on the final.

3. Write each of these statements in the form “if p then q” in English. • It is necessary to wash the boss’s car to get promoted • A sufficient condition for the warranty to be good is that you bought the computer less than a year

ago • Carol gets sea sick whenever she is on a boat

4. State the converse, the contrapositive and inverse of each of these statements:

• If it snows tonight, then I will stay at home. • I go to the beach whenever it is a sunny summer day.

5. Construct a truth table for each of these compound propositions

• (p ∧ q) ∨ r • p → (¬q ∨ r) • (p → q) ∧ (¬p → r) • ((p → q) → r)  → s

6. Show that the following statements are tautology without using truth tables:

• [¬p    ∧  (p  ∨  q)]  →  q • [p    ∧  (p  →  q)]  →  q

7. Show that

• p  →  q and ¬q  →  ¬p are logically equivalent. • (p  →  q)  ∨  (p  →  r)  and  p  →  (q  ∨  r)  are logically equivalent.

8. Translate these statements into English, where R(x) is “x is a rabbit” and H(x) is “x hops” and the domain consists of all animals.

• ∀x (R(x) → H(x)) • ∃ x (R(x) ∧ H(x))

Find the negation of the above statements. 9. Let Q(x) be the statement “x + 1 > 2x”. If the domain consists of all integers, what are the truth values of the following expressions? Q(0), Q(-1), ∃ xQ(x), ∀ xQ(x), ∃ x¬Q(x), ∀ x¬Q(x) 10. Let P(x, y) be the statement “student x has taken class y” where the domain of x consist of all students in your class and the domain of y consists of all computer engineering courses at your school. Express in English the following two quantifications and their negations. ∃ x ∃ y P(x, y) and ∃ x ∀y P(x, y) 11. Let F(x, y) be the statement “x can fool y” where the domain consists of all people in the world. Use quantifiers to express each of these statements.

• Everybody can fool Fred • Everybody can fool somebody • There is no one who can fool everybody • Everyone can be fooled by somebody • There is exactly one person whom everyone can fool.

12.

• Use a direct proof to show that the sum of two even integers is even. • Prove that if n is a perfect square, then n + 2 is not a perfect square. • Prove that if x is rational and x ≠ 0, the 1/x is rational. • Prove that is n is an integer and 3n+2 is even, then n is even • Prove that if n is a positive integer, then n is even if and only if 7n+4 is even. • Prove that m2 = n2 if and only if m = n or m = -n.