Homework #1: due Monday 1/10/2004

1. Which of the following expressions are statements? Circle the statements.

a) The US has 49 states.

b) I like to eat pizza and you often think of going to the movies.

c) Call me on Thursday if you are home.

d) If we go out tonight, the babysitter will be unhappy.

e) 4<3

f) 4+3

g) If x>2 then x³>1.

2. Let P=”I am happy”, Q=”I am watching a movie”, R=”I am studying for Math310”.

a) Translate the following statements into words:

i. not (P or Q)

ii. Q => not R

iii. (not P and R) or Q

b) Translate the following statements into symbols:

i. I am neither studying for Math 310, nor watching a movie.

ii. I am happy when I study for Math 310.

iii. I don’t study for Math310 if I am watching a movie.

c) If P and R are true, but Q is false, what are the truth values of the six statements from parts a) and b)?

3. Fill in the truth tables for the following statements:

P	Q	((P=>Q) and P )=> Q
T	T
T	F
F	T
F	F

P	Q	Not (P=>Q) => P
T	T
T	F
F	T
F	F

4. a) Give definitions for

i. Tautology (hint: statements such as those in problem 2.4)

ii. Contrapositive of an implication. (hint: see problem 2.5 (i))

(Note that 2.5 (i) shows that an implication is logically equivalent to its counterpositive)

b) Which of the following statements is a tautology?

i. If John eats an anchovy pizza, then he either eats an anchovy pizza or he does not.

ii. If John either eats an anchovy pizza or he does not, then he eats an anchovy pizza.

iii. If pigs have wings and pigs do not have wings, then the sun sets in the East.

c) State the contrapositive of the following statement: If it’s Tuesday, we must be in Belgium.

5. Prove that for any integer n, n² + n is even.
(Show all details and use the definition: an integer x is even if there exists an integer k such that x=2k.)

6. Prove that for all non-negative real numbers x and y, .