Consider the statement, “Whenever Holmes wears a purple shirt and the green vest, he chooses to not wear a bow tie.” Let \(P\) be the statement, “Holmes wears a purple shirt,” \(G\) be the statement, “Holmes wears the green vest,” and \(B\) be the statement, “Holmes wears a bow tie.” Which of the following is the best translation of the statement into propositional logic?
\((P \wedge G) \imp \neg B \)
\((P \wedge G) \imp B \)
\((P \vee G) \imp \neg B \)
\(P \wedge (G \imp B) \)
2.
Consider the statement, “Holmes never wears the green vest unless he is also wearing either a purple shirt or red shoes.” With \(P\) and \(G\) as in the previous question, and \(R\) being the statement, “Holmes wears red shoes,” which of the following is the best translation of the statement into propositional logic?
\(G \imp (P \vee R) \)
\(\neg G \imp (P \vee R) \)
Consider the case where Holmes does wear a green vest but does not wear a purple shirt or red shoes. That would make \(\neg G\) false and \(P \vee R\) true, so the implication would be true. But in this situation, the original statement would be false.
\((P \vee R) \imp G \)
Consider the case where Holmes does not wear a purple shirt or red shoes, and does wear the green vest. That would make \(P \vee R\) false and \(G\) true, so the implication would be true. But in this situation, the original statement would be false.
\((P \vee R) \imp \neg G \)
Consider the case where Holmes does not wear a purple shirt or red shoes, and does wear the green vest. That would make \(P \vee R\) false and \(\neg G\) false, so the implication would be true. But in this situation, the original statement would be false.
3.
Consider the statement, “If you major in math, then you will get a high-paying job,” and the statement, “Either you don’t major in math or you will get a high-paying job.” In which of the following cases are both statements true? Select all that apply.
You major in math and get a high-paying job.
You major in math and don’t get a high-paying job.
In fact, in this case, both of the statements are false.
You don’t major in math and do get a high-paying job.
This makes the implication true because the if part is false. The disjunction is true because the first part is true.
You don’t major in math and don’t get a high-paying job.
