The Foundations: Logic and Proofs – Exercises from 21 to 25

21. For each of these sentences, determine whether an inclusive or, or an exclusive or, is intended. Explain your answer.

a) Coffee or tea comes with dinner.

b) A password must have at least three digits or be at least eight characters long.

c) The prerequisite for the course is a course in number theory or a course in cryptography.

d) You can pay using U.S. dollars or euros.

Solution

a) Exclusive or. The diner gets to choose only one of the beverages so this is exclusive or.

b) Inclusive or. Long passwords with many digits are acceptable so, inclusive or.

c) Inclusive or. The student may have both the courses so, inclusive or.

d) Inclusive or. The patron would pay a portion of the bill in dollars and the remainder in euros, inclusive or.

22. For each of these sentences, determine whether an inclusive or, or an exclusive or, is intended. Explain your answer.

a) Experience with C++ or Java is required.

b) Lunch includes soup or salad.

c) To enter the country you need a passport or a voter registration card.

d) Publish or perish.

Solution

a) Inclusive or. Experience with C++ or Java is required: “Inclusive or” (Having experience in both fields would be even beter but what the scentence means is that having experience with at least one is necesary.)

b) Exclusive or. Lunch includes soup or salad: “Exclusive or intended” (The lunch comes with either soup or salad but never with soup and salad.)

c) Inclusive or. To enter the country you need a passport or a voter’s registration card. “Inclusive or intended” (Having both would be even better but what the sentence means is that having least one is necessary to enter the country.)

d) Exclusive or. Publish or perish. “Exclusive or intended” (You either publish or perish but you can’t perish and publish.)

23. For each of these sentences, state what the sentence means if the logical connective or is an inclusive or (that is, a disjunction) versus an exclusive or. Which of these meanings of or do you think is intended?

a) To take discrete mathematics, you must have taken calculus or a course in computer science.

b) When you buy a newcar from Acme Motor Company, you get $2000 back in cash or a 2% car loan.

c) Dinner for two includes two items from column A or three items from column B.

d) School is closed if more than two feet of snow falls or if the wind chill is below −100 ◦F.

Solution

a) Inclusive. Someone that has taken calculus and a course in computer science would be better prepared to take discrete mathematics than someone who has only taken one or the other.

b) Exclusive. A car comparw is more likely to give one incentive rather than both since it would reduce their

c) Exclusive. A restaurant would not be too keen on the idea of having you pay so little for choosing five items when you were only supposed to pick two from column A, or three from column B. Think Applebee’s

d) Inclusive. It would be absurd if a school didn’t close when more than 2 feet of snow falls and the wind chill is below 100, when either one of those results in a school closure.

24. Write each of these statements in the form “if p,then q” in English. [Hint: Refer to the list of common ways to express conditional statements provided in this section.]

a) It is necessary to wash the boss’s car to get promoted.

b) Winds from the south imply a spring thaw.

c) A sufficient condition for the warranty to be good is that you bought the computer less than a year ago.

d) Willy gets caught whenever he cheats.

e) You can access the website only if you pay a subscription fee.

f) Getting elected follows from knowing the right people.

g) Carol gets seasick whenever she is on a boat.

Solution

a) “q is necessary for p” or (“it is neccessary to q to p”) is a common way to re-express ” If p, then q”. Thus we can then rewrite the given sentence as:

If got promoted, thon you washed the boss’s car.

b) “p implies q” is a common way to re-express “If p, then q”. Thus can then rewrite the giwn sentence as:

If the winds come from the south, then there will be a spring thaw.

c) “A sufficient condition for q is p” is a common way to re-express “If p, then q”. Thus we can then rewrite the given sentence as:

If you bought tho computer less than a year ago , then the warranty is good.

d) ” q whenever p” is a common way to re-express “If p, then q”. Thus can then rewrite the given sentence as:

If Willy cheats, then he gets caught.

e) “p only if q” is a common way to re-express “If p, then q” , Thus we can then rewrite the given sentence as:

If can access the website, then you pay a subscription fee.

f) “q follows from p” is a common way to re-express “If p, then q”. Thus we can then rewrite the given sentence as:

If know tho right people, then you get elected.

g) “q whenever p” is a common to re-express “If p, then q”. Thus we can then rewrite the given sentence as:

If Carol got  on a boat, then she gets seasick.

25. Write each of these statements in the form “if p,then q” in English. [Hint: Refer to the list of common ways to express conditional statements.]

a) It snows whenever the wind blows fromthe northeast.

b) The apple trees will bloom if it stays warm for a week.

c) That the Pistons win the championship implies that they beat the Lakers.

d) It is necessary to walk eight miles to get to the top of Long’s Peak.

e) To get tenure as a professor, it is sufficient to be world famous.

f) If you drive more than 400 miles, you will need to buy gasoline.

g) Your guarantee is good only if you bought your CD player less than 90 days ago.

h) Jan will go swimming unless the water is too cold.

i) We will have a future, provided that people believe in science.

Solution

a) If the wind blows from the northeast, then it snows.

b) If it stays warm for a week, then the apple trees will bloom.

c) If the Pistons win the championship, then they beat the Lakers.

d) If you get to the top of Longs Peaks, then you must have walked 8 miles

e) If you are world famous, then you will get tenure as a professor.

f) If you drive more than 400 miles, then you will need to buy gasoline.

g) If your guarantee is good , then you must have bought your CD player less than 90 days

h) If the water is not too cold, then Jan will go swimming.

i) If people believe in science, then we will have a future.