Vfrdtyky

Just wondering what other ways $neg p to q$ can be expressed.

I know that $pto q$ is logically equivalent to $neg plor q$, hence I think that $neg pto q$ has the same logical equivalence as $plor q$.

As others have pointed out, truth tables will confirm the equivalence of $neg p to q$ and $p lor q$.

But also you'll get additional insight by thinking informally. Suppose you are given that either $p$ or $q$. Then, if you rule out $p$, then you are left with $q$. So, in symbols, from $p lor q$ you can infer $neg p to q$.

Conversely, suppose you are given that if not-$p$ then $q$. Then assuming you accept excluded middle, the principle that either $p$ or not $p$, it follows that either $p$ or (using that conditional) $q$. So, in symbols, from $neg p to q$ you can infer $p lor q$.

You have pretty much given the answer yourself already. Logical equivalence of $p$ and $q$ is given if $p$ is true if and only if $q$ is true (and hence $p$ is false iff $q$ is false). Since $p$ and $q$ can only be true or false, you can use truth tables and check whether logical equivalence is given. From there you can derive permissible manipulations of propositions.

In this case, $(lnot p) rightarrow q equiv lnot(lnot p) lor q equiv p lor q$, as you stated correctly.

As you said, $pRightarrow q$ is logically equivalent to $neg pvee q$.
Then by double negation, $neg p Rightarrow q$ is logically equivalent to $neg(neg p)vee q$, which amounts to $pvee q$.