The Foundations: Logic and Proofs – Exercises from 16 to 20

16. 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.

b) You get an A on the final, you do every exercise in this book, and you get an A in this class.

c) To get an A in this class, it is necessary for you to get an A on the final.

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.

e) Getting an A on the final and doing every exercise in this book is sufficient for getting an A in this class.

f) 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.

Solution

\\ a) r \wedge ¬q \\ b) p \wedge q \wedge r \\ c) r \Rightarrow p \\ d) p \wedge ¬q \wedge r \\ e) (p \wedge q) \Rightarrow r \\ f) r \Leftrightarrow (q \vee p)

17. Let p, q,and r be the propositions

p: Grizzly bears have been seen in the area.

q: Hiking is safe on the trail.

r: Berries are ripe along the trail.

Write these propositions using p, q,and r and logical connectives (including negations).

a) Berries are ripe along the trail, but grizzly bears have not been seen in the area.

b) Grizzly bears have not been seen in the area and hiking on the trail is safe, but berries are ripe along the trail.

c) If berries are ripe along the trail, hiking is safe if and only if grizzly bears have not been seen in the area.

d) It is not safe to hike on the trail, but grizzly bears have not been seen in the area and the berries along the trail are ripe.

e) For hiking on the trail to be safe, it is necessary but not sufficient that berries not be ripe along the trail and for grizzly bears not to have been seen in the area.

f) Hiking is not safe on the trail whenever grizzly bears have been seen in the area and berries are ripe along the trail.

Solution

\\ a) r \wedge -p \\ b) -p \wedge q \wedge r \\ c) r \Rightarrow (q \Leftrightarrow -p) \\ d) -q \wedge -p \wedge r \\ e) (q \Rightarrow (-r \wedge p)) \wedge -((-r \wedge -p) \Rightarrow q) \\ f) (p \wedge r) \Rightarrow -q

18. Determine whether these biconditionals are true or false.

a) 2 + 2 = 4 if and only if 1 + 1 = 2.

b) 1 + 1 = 2 if and only if 2 + 3 = 4.

c) 1 + 1 = 3 if and only if monkeys can fly.

d) 0 > 1 if and only if 2 > 1.

Solution

a) True. 2 + 2 = 4 is true, also 1 + 1 = 2 is true. Therefore, the compound proposition is true.

b) False. 1 + 1 = 2 is true, however 2 + 3 = 4 is false. Therefore, the compound proposition is false.

c) True. 1 + 1 = 3 is false, also, monkeys can fly, is false. Therefore, the compound proposition is true.

d) False. 0 > 1 is false, however 2 > 1 is true. Therefore, the compound proposition is false.

19. Determine whether each of these conditional statements is true or false.

a) If 1 + 1 = 2, then 2 + 2 = 5.

b) If 1 + 1 = 3, then 2 + 2 = 4.

c) If 1 + 1 = 3, then 2 + 2 = 5.

d) If monkeys can fly, then 1 + 1 = 3.

Solution

a) False. This biconditonal statement is false because p is true and q is false.

b) True. This conditional statement is true because p is false, but q is true, making it true.

c) True. This is a conditional statement. Since p and q are both false, this makes the answer true.

d) True. This is a conditional statement. Since they a both false, then the answer is true.

20. Determine whether each of these conditional statements is true or false.

a) If 1 + 1 = 3, then unicorns exist.

b) If 1 + 1 = 3, then dogs can fly.

c) If 1 + 1 = 2, then dogs can fly.

d) If 2 + 2 = 4, then 1 + 2 = 3.

Solution

a) True. In this conditional statement p and q are both false, so the answer is true.

b) True. In this conditional statement p and q are both false, so the answer is true.

c) False. In this conditional statement, since p is true and q is false, the statement will be false.

d) False. Both statements in this conditional are true, so the answer is true.