Truth Tables and Implications Query

Question

$$ \begin{array}{ | m{5em} | m{1cm}| m{1cm} | } \hline p& q & r & (p\vee \neg q)\to r \\ \hline F & F & F & F \\ \hline F & F & T & T \\ \hline F & T & F & T \\ \hline F & T & T & T \\ \hline T & F & F & F \\ \hline T & F & T & T \\ \hline T & T & F & F \\ \hline T & T & T & T \\ \hline \end{array}$$ I am having some difficulty understanding this truth table. What I don't understand is that, how can $F~~T~~F$ be true, yet $F~~F~~F$ is false?

Any explanations will be appreciated :)

Answer 1

If $p$, $q$ and $r$ are all false, $p\vee \neg q = F \vee T$ is true. But $T \to F$ is false, so the total expression is false.

If instead $q$ is true, $p\vee \neg q = F \vee F$ is false. Because $F \to F$ is true, the total expression is true.

Answer 2

Case FTF -

$p\vee \neg q$

$F \vee F = F$

Now,

$(p\vee \neg q)\to r$

$F \to F = T$

As false can implies false.

Case FFF -

$p\vee \neg q$

$F \vee T = T$

Now,

$(p\vee \neg q)\to r$

$T \to F = F$

As true cannot implies false.