Is the following sentence a tautology: $(p\Rightarrow q)\vee(r \Rightarrow p)\vee(r\Rightarrow s)\vee(r\Rightarrow q)$?

Question

If both $p$ and $q$ are false then ($p\Rightarrow q$) is true. If either $p$ or $q$ is true then one of ($r\Rightarrow p$) or ($r\Rightarrow q$) is true. If both $p$ and $q$ are true then all are true. Since they are OR'ed, one way of an other is true. IT IS A TAUTOLOGY..

My prof dint accept my answer, he said need little more consideration.

Answer 1

Your professor is mistaken, and your argument is correct, though it could be simplified slightly without any major change, since you could combine the second and third cases:

If $p$ and $q$ are both false, then $p\to q$ is true; otherwise at least one of $p$ and $q$ is true, and therefore at least one of $r\to p$ and $r\to q$ is true. In all cases, therefore, at least one of the implications is true and therefore their disjunction is true.

However, it can be simplified even more by looking just at $p$: if $p$ is true, then $r\to p$ is true, and if $p$ is false, then $p\to q$ is true.