Q:

Deducing formula from a given set of formulas

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:

¬p ∨ r
¬p ∨ p
¬p
s
s ∨ ¬p ∨ q

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

A:

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 
$\neg q \lor s \lor r$

