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Consider the statement:
If Bill takes Sam to the concert, then Sam will take Bill to dinner.
Which of the following implies that this statement is true?
Alright.
a. Sam takes Bill to dinner only if Bill takes Sam to the concert.
___Equivalent to Q-->P, which is true if Q is false and P is true, which would make P-->Q false. So this does not imply the above statement is true.
Agree. Or more succinctly, for $Q = \F$ and $P=\T$, $(Q \rightarrow P) \rightarrow (P \rightarrow Q) = \F \rightarrow \T = \F$
b. Either Bill doesn't take Sam to the concert or Sam takes Bill to dinner.
___If Bill doesn't take Sam to the concert, then P is false and regardless of Q, P-->Q is true. If Sam takes bill to dinner, then Q is true and regardless of P, P-->Q is true. So this implies the above statement is true.
$$(\lnot P \xor Q) \rightarrow (P \rightarrow Q)$$
Correct, either case of the condition being true makes the resultant true.
c. Bill takes Sam to the concert.
___Then P is true. But if Q is F, the statement is false, so this does not imply the above statement is true.
$$P \rightarrow (P \rightarrow Q)$$
Correct again. $P \and \lnot Q$ is a counterexample.
d. Bill takes Sam to the concert and Sam takes Bill to dinner.
___Equivalent to P/\Q, and which does imply the above statement to be true.
$$(P \and Q) \rightarrow (P \rightarrow Q)$$
Correct again.
e. Bill takes Sam to the concert and Sam doesn't take Bill to dinner.
___Equivalent to P/\~Q, so this does not imply the above statement is true.
$$(P \and \lnot Q) \rightarrow (P \rightarrow Q)$$
Correct again.
f. The concert is canceled.
___Then P is always false, so P-->Q is true regardless of truth value for Q. This statement implies the above to be true.
$$\lnot P \rightarrow (P \rightarrow Q)$$
Correct.
g. Sam doesn't attend the concert.
___Equivalent to P being false, so P-->Q is true regardless of truth value for Q. This statement implies the above to be true.
$$\lnot P \rightarrow (P \rightarrow Q)$$
Correct.
They all seem correct to me.
Consider the sentence:
I will jump off the cliff if and only if you do it as well
You did not jump off the cliff - so why should I?
The moral of the story is that a biconditional statements only states that $\alpha$ holds whenever $\beta$ is the case, they are, say, 'logically attached'. In the cliff analogy, they either jump 'together' or not.
Best Answer
It is false.
Consider when both $p$ and $q$ are true. Then the RHS is true, whereas, since $p\implies q$ is true, the LHS is false.