The Foundations: Logic and Proofs – Exercises from 6 to 10

6. What is the negation of each of these propositions?

 

a) Jennifer and Teja are friends.

b) There are 13 items in a baker’s dozen.

c) Abby sent more than 100 text messages yesterday.

d) 121 is a perfect square.

Answer:

 

a) Jennifer and Teja are not friends.

The opposite of a statement can be obtained by adding “not” to the given statement. The opposite of the two people being friends is that they are not friends.

b) There are not 13 items in a baker’s dozen.

The opposite of a statement can be obtained by adding “not” to the given statement: There are not 13 items in a baker’s dozen.

c) Abby sent at most 100 text messages ewery day.

The opposite of “more than” is “at most” (or “less than or equal to”): Abby sent at most 100 text messages ewery day.

d) 121 is not a perfect square.

The opposite of a statement can be obtained by adding “not” to the given statement: 121 is not a perfect square.

7. What is the negation of each of these propositions?

 

a) Steve has more than 100 GB free disk space on his laptop.

b) Zach blocks e-mails and texts from Jennifer.

c) 7 11 13 = 999.

d) Diane rode her bicycle 100 miles on Sunday.

Answer:

 

a) It is not the case that Steve has more than 100 GB free space on his laptop.

b) It is not the case that Zach blocks e-mails and texts from Jennifer.

c) 7.11.13 ≠ 999

d) It is not the case that Diane rode her bike 100 miles on Sunday.

8. Suppose that Smartphone A has 256 MB RAM and 32 GB ROM, and the resolution of its camera is 8 MP; Smartphone B has 288 MB RAM and 64 GB ROM, and the resolution of its camera is 4 MP; and Smartphone C has 128 MB RAM and 32 GB ROM, and the resolution of its camera is 5 MP. Determine the truth value of each of these propositions.

 

a) Smartphone B has the most RAM of these three smartphones.

b) Smartphone C has more ROM or a higher resolution camera than Smartphone B.

c) Smartphone B has more RAM, more ROM, and a higher resolution camera than Smartphone A.

d) If Smartphone B has more RAMand more ROM than Smartphone C, then it also has a higher resolution camera.

e) Smartphone A has more RAM than Smartphone B if and only if Smartphone B hasmore RAMthan Smart phone A.

Answer:

 

a) True

Smartphone B has 288 MB RAM, which is the most out of the other two phones.

b) True

Either the ROM has to be greater or the resolution has to be greater. The resolution is greater in this example so it is true.

c) False

It is false because the resolution is larger in Smartphone A.

d) False

It may have more RAM and ROM, but Smartphone B’s resolution is less making the
statement false.

e) False

It is a biconditional statement which is F – T making the statement false.

9. Suppose that during the most recent fiscal year, the annual revenue of Acme Computer was 138 billion dollars and its net profit was 8 billion dollars, the annual revenue of Nadir Software was 87 billion dollars and its net profit was 5 billion dollars, and the annual revenue of Quixote Media was 111 billion dollars and its net profit was 13 billion dollars. Determine the truth value of each of these propositions for the most recent fiscal year.

 

a) Quixote Media had the largest annual revenue.

b) Nadir Software had the lowest net profit and Acme Computer had the largest annual revenue.

c) Acme Computer had the largest net profit or Quixote Media had the largest net profit.

d) If Quixote Media had the smallest net profit, then Acme Computer had the largest annual revenue.

e) Nadir Software had the smallest net profit if and only if Acme Computer had the largest annual revenue.

Answer:

 

a) False

This proposition is false because Quixote Media had less annual revenue than Acme Computer ($111 billion < $138 billion).

b) True

This proposition is true because both statements are true.

c) True

This proposition is true because Quixote Media had the largest net profit. Note: Since this an v statement, either statement being true the proposition true.

d) True

This proposition is true because Acme Computer had the largest rewnue. Note: this is a proposition.

Even though Quixote did not haw the smallest net profit (hypothesis), the conclusion still holds.

e) True

This proposition is true because in both directions the conclusions hold true.

10. 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

b) p q

c) p → q

d) p q

e) p ↔ q

f) ¬p → ¬q

g) ¬p ¬q

h) ¬p (p q)

Answer:

 

a) I did not buy a lottery ticket this week.

b) I bought a lottery ticket this week or I won the million dollar jackpot.

c) If I bought a lottery ticket this week, then I won the million dollar jackpot.

d) I bought a lottery ticket this week and I won the million dollar jackpot.

e) I bought a lottery ticket this week if and only if I won the million dollar jackpot.

f) If I did not buy a lottery ticket this week, then I did not win the million dollar jackpot.

g) I did not buy a lottery ticket this week and I did not win the million dollar jackpot.

h) I did not buy a lottery ticket this week or, I bought a lottery ticket this week and won the million dollar jackpot.bes