Here is the truth table for an implication:
$$ \begin{array}{ccccc}
P & & Q & & P \to Q \\
T & & T & & T \\
T & & F & & F \\
F & & T & & T \\
F & & F & & T \end{array} $$
You can think of an implication as a conditional promise. If you keep the promise, it's true. If you break the promise, it's false.
If I tell my kids, "I'll give you a cookie if you clean up." Then they clean up. I better give them a cookie. If I don't, I've lied. However, if they don't clean up, I can either give them a cookie or not. I didn't promise either if they didn't keep up their end of the bargain.
So in other words, an implication is false only if the hypothesis is true and conclusion is false.
Logically $P \to Q$ is equivalent to $\neg P \vee Q$
I had a computer science professor who was fond of promising his kids things given a false premise. This way he wasn't compelled to follow through. Example: "If the moon is made of green cheese, I'll give you an x-box."
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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.
Best Answer
Perhaps this will help. This can be seen as a way of explaining modus ponens with which you may be more familiar:
If we know $p$ implies $q$ (is true), and $p$ happens to be true, then $q$ must be true:
$$p \rightarrow q$$ $$p$$ $$\therefore q$$
$p \rightarrow q,$ by itself, tells us nothing about the truth value of $q$. IF $p\rightarrow q$ is true we know that either $p$ is false, or $q$ is true, or both, which is nicely expressed as follows: $$(p \rightarrow q) \iff (\lnot p \lor q)$$