Why is the last variable in this truth table (the bottom right) false?

Question

Table seen here.

It was my understanding that the conditional statement (->) in truth tables was only false in circumstances where the variables you are comparing are TF. However, when comparing ~pvq to r the last line is FF yet most sources that I go to label it as false. Is there another caveat that I don't know about?

Thanks for any help!

Answer 1

In the last line,

$$\lnot p \lor q = \lnot F \lor F = T \lor F = T$$

Thus

$$\lnot p \lor q \to r = T \to F$$ is false.

Answer 2

Hint: $$\lnot p\lor q \to r =\lnot(\lnot p+q)+r=pq'+r $$ Whenever $\lnot p+q$ is false or $r$ is true or both conditions hold then the expression is true.

Answer 3

It might be helpful to note that $p \rightarrow q$ is equivalent to $\lnot p \vee q$. In an implication, either the premise fails or the conclusion holds.

Applying this identity to $\lnot p \vee q \rightarrow r$ yields $(p \wedge \lnot q) \vee r$, which evaluates to false in the instance you are asking about.