Deducing formula from a given set of formulas

Question

Having a set of formulas: $$ S = \big\{ p ⇒ (r \wedge p), \neg p \wedge s, s \vee (p ⇒ q)\big\} $$ I have to find a formula that can be deduced from this set of formulas. The answer we are looking for (according to the textbook) is $$ \neg q \vee s \vee r $$ Could anyone give me advice on how to solve this type of problem? How to deduce a formula? I have found some advice about writing the set in a CNF and then trying to resolve it, however I'm stuck with the rewritten formulas:

1. ¬p ∨ r
2. ¬p ∨ p
3. ¬p
4. s
5. s ∨ ¬p ∨ q

Where 1. and 2. comes from S[0], 3. and 4. from S[1] and 5. from S[2].

Answer 1

If your Formal System's Rules of Inferences allows Conjunction Elimination and Disjunction Introduction, you may deduce several well-formed formulas from your starting set, including that which your textbook describes.

Expanding on Bram28's answer, starting from $\neg p \land s$, use Conjunction elimination to deduce $s$.

$\neg p \land s \vdash s$

From there, you can use Disjunction Introduction to introduce the required disjuncts (and others). E.g.

$s \vdash s \lor \neg q$

and then

$s \lor \neg q \vdash s \lor \neg q \lor r$

which you can rearrange because of associativity of $\lor$ into

$\neg q \lor s \lor r$

Answer 2

If $s$ is true, then $\neg q \lor s \lor r$ is true as well.