I can't wrap my head around those validity problems, especially this one:
$ Establish \ the \ validity \ of \ the \ following \ argument: $
$(\lnot p \lor q) \to r \\ r \to (s \lor t) \\ \lnot s \land \lnot u \\ \lnot u \to \lnot t \\ \therefore p $
What I have achieved so far:
$p \to q \to r \\ r \to (s \lor t) \\ \lnot s \land \lnot u \\ \lnot u \to \lnot t \\ \therefore \\ p \to q \to r \\ r \to \lnot s \to t \\ \lnot u \\ \lnot u \to \lnot t \\ \therefore \\ p \to t \\ \lnot t \\ \therefore \\ \lnot p$
and I've achieved a different result by solving the problem with a different approach but what I've mentioned is a lot simpler
$$\neg s \land \neg u\\\therefore \neg s \\ \neg u$$since $\neg u\to \neg t$ therefore $\neg t$ is true and because of $r\to (s\lor t)$ we have$$\neg (s\lor t)=\neg s\land \neg t \to \neg r$$and $\neg s$ and $\neg t$ are true so is $\neg r$. Also $\neg p \lor q \to r$ therefore $$\neg r\to \neg(\neg p \lor q)=p\land \neg q$$ and $\neg r$ is true so we can conclude $p\land\neg q$ is true and both $p$ and $\neg q$ are true and we have proven what we wanted