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.
Table of Contents
- Definitions
- Information Summary
- 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.
- Summary
Let’s start with the definitions we will need to solve this exercise.
Definitions
Because the exercise involves compound propositions, we need to have at hand the truth values for compound propositions.
See the figure below.
It is very useful for me to summarize the information we get from the exercise. In doing that, we are going to save time and mistakes in the future.
Information Summary
| Acme Computer | Nadir Software | Quixote Media | |
| Annual revenue | 138 billion dollars | 87 billion dollars | 111 billion dollars |
| Net profit | 8 billion dollars | 5 billion dollars | 13 billion dollars |
Now we are ready to answer each question.
a) Quixote Media had the largest annual revenue
From the table above we can see that Acme Computer had the larger annual revenue. So, we can state that the truth value, in this case, is false.
Answer: The truth value of the proposition is false.
b) Nadir Software had the lowest net profit and Acme Computer had the largest annual revenue.
From our table for the truth values in the definitions section, we know that for a compound proposition p^q to be true, both p and q must be true.
So, from the information summary, we can see that Nadir software had the lowest net profit (5 billion). Also, Acme Computer had the largest annual revenue.
Because both are true, we know that the conjunction (^) of both propositions is also true (according to the truth values of compound propositions, see table in the definition section).
Answer: The truth value of the proposition is true.
c) Acme Computer had the largest net profit or Quixote Media had the largest net profit.
Let’s first analyze the facts according to our information summary table.
Acme Computer didn’t have the largest net profit (8 billion). So, the proposition “Acme Computer had the largest net profit” is false.
Quixote Media had the largest net profit (13 billion). In this case, the proposition is true.
If we look at the table of truth values of compound propositions (definitions section above), we can see that “p v q” will be true if p, or q, or both are true, in other words, if one of the two is true. In this case, one of them is true. Therefore, we can state that the compound proposition is true.
Answer: The truth value of the proposition is true.
d) If Quixote Media had the smallest net profit, then Acme Computer had the largest annual revenue.
Again, let’s analyze the facts according to our information summary:
- Quixote Media had the smallest net profit. No, it has the largest net profit. Therefore, this proposition is false.
- Acme Computer had the largest annual revenue. True, 138 is the largest annual revenue among the three companies.
Now, let’s got to the table in the definitions section.
p-> q is true in three cases:
- p is true and q is true
- p is false and q is true
- both p and q are false
In our case, we can state that p = “Quixote Media had the smallest net profit” and q = “Acme Computer had the largest annual revenue”.
So, p is false, and q is true. This is one of the cases where the compound proposition (->) is true. So, we got our answer.
Answer: The truth value of the proposition is true.
e) Nadir Software had the smallest net profit if and only if Acme Computer had the largest annual revenue.
Let’s start analyzing the facts according to our information summary:
- Nadir Software had the smallest net profit. This is true, 5 billion is less than 8 and 13.
- Acme Computer had the largest annual revenue. Also true, 138 is the largest profit of the three companies.
Now, we should go to the table that has the truth values of compound propositions
The part of the sentence “if and only if” tells us this is the biconditional. This compound proposition (p <-> q) will be true only when both p and q have the same value. In other words, both p and q are true, or both p and q are false.
In this case, we already stated that both propositions are true.
Answer: The truth value of the proposition is true.
Summary
In this post, I gave you several ways of analyzing this type of exercise. You should use the one that works better for you.
Summarizing information from the problem helps you keep organized and it will save time when you are answering different questions.
Always have the definitions at hand. In this case, what you needed was the truth values of compound propositions.
Related posts:
- How to create the truth table for a compound proposition
- Propositional Logic: Exercise 6
- What is the negation of each of these propositions?
- What are propositional equivalences in Discrete Mathematics?
- Which of these sentences are propositions? What are the truth values of those that are propositions?
- Propositional logic Exercise 11. Write these propositions using p and q and logical connectives (including negations)
- 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?
