Solution

There are three variables $p,q,r$.

So, total number of ordered tuples is:

$2^3=8$

The implication $A\Rightarrow r$ is false only when $A$ is true and $r$ is false.

Here,
$A=(\neg p\vee q)$

Now, $A=(\neg p\vee q)$ is false only when both $\neg p$ and $q$ are false.

That means:

$p$ is true and $q$ is false.

So, $A$ is false in only $1$ case out of $4$ possible cases of $(p,q)$.

Therefore, $A$ is true in $3$ cases.

For implication to be false, $A$ must be true and $r$ must be false.

So, false cases $=3$

Hence, true cases:

$8-3=5$