I have a problem with an exercise of the Chapter 2 (Logic) of the book Mathemathical Proof: A transition to advanced mathematics.
It says:
Consider the statement (implication):If Bill takes Sam to the concert, then Sam will take Bill to dinner. Which of the following implies that this statement is true?
(a) Sam takes Bill to dinner only if Bill takes Sam to the concert.
(b) Either Bill doesn’t take Sam to the concert or Sam takes Bill to dinner.
(c) Bill takes Sam to the concert.
(d) Bill takes Sam to the concert and Sam takes Bill to dinner.
(e) Bill takes Sam to the concert and Sam doesn’t take Bill to dinner.
(f) The concert is canceled.
(g) Sam doesn’t attend the concert.
The answer in the solution manual of the book is the following: 2.28 (b), (d), (f), (g) are true.
Why (f) and (g) are solutions? They say nothing about the truth of the proposition, right? Moreover, I don't know why (b) (P∨Q) implies the truth of the statement implication (P⇒Q). Because if P is true and Q is false, P∨Q is true, but P⇒Q is not.
In exchange, I see (d) is correct. My solution would be (d).
I may be making some mistake. I'd like you to point it out or to clarify if my solution is right. Thank you in advance.
The given conditional statement is false only in the case that
“Bill takes Sam to the concert” is true, and
“Sam will take Bill to dinner” is false.
In another combination of the truth values of these two clauses, the conditional statement is true.
In case that “the concert is canceled”, the clause (1) becomes false, then the conditional statement is true regardless of whether “Sam will take Bill to dinner” or not.
Likewise, for the clause “Sam doesn’t attend the concert” which renders (1) false.