How can this be a tautology?

Question

The formula:

$r ∧ q ∧ ¬ p ∨ r ∧ p ∧ ¬ q ⇒ r ∨ ¬ p ∨ ¬ q$

The answer says it's a tautology but I've tried to work it out and I get false.

My truth table to the first half of the formula is in the photo and you can see that I got false so where have I gone wrong? Image

I've done the right hand side

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So because the right side is true, it's a tautology?

Answer 1

In your truth table you only worked out the truth-conditions of $(r \land q \land \neg p) \lor (r \land p \land \neg q)$. Now you need to work out $r \lor \neg p \lor \neg q$, and then use the $\Rightarrow$ to get the truth-conditions of the whole conditional.

Remember that the $\Rightarrow$ is false when the antecedent is true and the consequent is false, and is true otherwise. So, in all those cases where $(r \land q \land \neg p) \lor (r \land p \land \neg q)$ evaluates to false, the whole conditional is automatically true, which covers $6$ out of the $8$ rows. So, you just need to check that in the other $2$ rows, $r \lor \neg p \lor \neg q$ is indeed true.

Answer 2

You have gone wrong nowhere. It is a very common misconception among freshmen. An implication is false only when the antecedent is true and the consequent is false. For instance, the sentence "if it rains tomorrow, I'll carry an umbrella" is false only if it DOES rain and you DON'T carry an umbrella. If it doesn't rain, you've spoken truth, whether you carry an umbrella or not. It's also true if it does rain and, as a man or woman of your word, you do bring along an umbrella.

So, all those cases where you've gotten a false truth value for the antecedent? They straightaway mean that the full sentence is true! You need only examine whether the consequent is also true in those two other cases, and you'll have proven indeed that the above sentence is a tautology.

I'm sure you didn't doubt that anyway - after all, the antecedent implies that r is true, which is sufficient to deduce the consequent.