Use basic inference rules to establish the validity of the argument. p → ¬q q ∨ r p ∨ u ¬r

To establish the validity of the argument, we can use basic inference rules, including Modus Ponens, Modus Tollens, Disjunctive Syllogism, and Addition.

Given:
1. p \rightarrow \neg q
2. q \lor r
3. p \lor u
4. \neg r

From p \rightarrow \neg q and p \lor u, we can use Disjunctive Syllogism to derive \neg p \lor \neg q.

Using Disjunctive Syllogism:
5. ¬p ∨ ¬q (from 1 and 3)

Now, from ¬p ∨ ¬q and \neg r, we can use Modus Tollens to derive \neg p.

Using Modus Tollens:
6. \neg p (from 5 and 4)

Next, from \neg p and p \rightarrow \neg q, we can use Modus Tollens to derive \neg q.

Using Modus Tollens:
7. \neg q (from 6 and 1)

Now, from \neg q and q \lor r, we can use Disjunctive Syllogism to derive r.

Using Disjunctive Syllogism:
8. r (from 7 and 2)

Therefore, the argument is valid, and the conclusion is r.

\boxed{r}