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?

/ Propositional Logic / By

As usual, we should start with the appropriate definitions. Remember that in mathematics, we must know the definitions before we can start trying to solve a problem.

Definitions

Let p and q be propositions. The disjunction of p and q, denoted by p ∨ q, is the proposition “p or q.” The disjunction p ∨ q is false when both p and q are false and is true otherwise.

Let p and q be propositions. The exclusive or of p and q, denoted by p ⊕ q, is the proposition that is true when exactly one of p and q is true and is false otherwise.

All the definitions I use are from the book Discrete Mathematics and its Applications by Rosen.

Now that we know the differences, according to the definitions, between the inclusive or and the exclusive or, we can start answering the questions.

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

Inclusive or: It means that you could take discrete mathematics if you took calculus, or a course in computer science, or both.

Exclusive or: It means that you can only take discrete mathematics if you took either calculus or a course in computer science.

In this case, I think the inclusive or was intended.

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

Inclusive or: It means that if you buy a new car from Acme, you can get $2000 cash back, or a 2% car loan, or both.

Exclusive or: If you buy a car, you can get either $2000 cash back or a 2% car loan, but not both.

In this case, I think the exclusive or was intended.

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

Inclusive or: It means that a dinner for two, includes two items from column A, or three items from column B, or five items two from column A and three from column B.

Exclusive or: It means that a dinner for two includes either two items from column A or three items from column B.

In this case, I think the exclusive or was intended.

d)School is closed if more than 2 feet of snow falls or if the wind chill is below −100

Inclusive or: It means that the school will be closed if any of the two conditions are met, or both.

Exclusive or: The school will be closed if more than 2 feet of snow falls, or if the wind chill is below -100, but not if both conditions are met.

In this case, I think the exclusive or was intended.

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