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.
$\\$
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.
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.
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.
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.
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.
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.
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.
$\\$
Can anyone confirm my answers?
Best Answer
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Alright.
Agree. Or more succinctly, for $Q = \F$ and $P=\T$, $(Q \rightarrow P) \rightarrow (P \rightarrow Q) = \F \rightarrow \T = \F$
$$(\lnot P \xor Q) \rightarrow (P \rightarrow Q)$$
Correct, either case of the condition being true makes the resultant true.
$$P \rightarrow (P \rightarrow Q)$$
Correct again. $P \and \lnot Q$ is a counterexample.
$$(P \and Q) \rightarrow (P \rightarrow Q)$$
Correct again.
$$(P \and \lnot Q) \rightarrow (P \rightarrow Q)$$
Correct again.
$$\lnot P \rightarrow (P \rightarrow Q)$$
Correct.
$$\lnot P \rightarrow (P \rightarrow Q)$$
Correct.
They all seem correct to me.