Using Rule of Inference, How to derive following conclusion from given premises?

Question

Question is from the book: Discrete Mathematical Structures with Applications to CS by Tremblay and Manohar. It is an exercise problem. But, unfortunately, there is no help available on answers, or solutions of this book. I have tried to solve this problem but couldn't get the desired conclusion.

Premise 1: $P \rightarrow Q$,

Premise 2: $(\neg Q \lor R) \wedge \neg R$

Premise 3: $ \neg (\neg P \wedge S)$

Conclusion: $ \neg S$

Solution:

1. $(\neg Q \lor R)$ $\wedge$ $-R$..............[Introducing Premise 2]

2. $(\neg Q \lor R)$.........................[Tautologically Implies, 1, Simplification]

3. $Q \rightarrow R$.............................[Tautologically Implies, 2, Converting Disjuction To Implication]

4. $P \rightarrow Q$.............................[Introducing Premise 1]

5. $P \rightarrow R$.............................[Tautologically Implies, 4, 3, Transitivity Law]

6. $ \neg (\neg P \wedge S)$.......................[Introducing Premise 3]

7. $P \lor \neg S$............................[Tautologically Implies, 6, DeMorgan's Law]

8. $ \neg S \lor P$............................[Tautologically Implies, 7, Commutative Law]

9. $S \rightarrow P$............................[Tautologically Implies, 8, Converting Disjuction to Implication]

10. $S \rightarrow R$............................[Tautologically Implies, 9, 5, Transitivity]

11. $\neg S \lor R$............................[Tautologically Implies, 10, Converting Implication to Disjuction]

What wrong I did? I am getting $\neg S \lor R$ instead of $\neg S $

Answer 1

hint

From premisse $ 2$, use distributivity to get $ \lnot Q \wedge \lnot R$ because $ R\wedge \lnot R $ is false.

by simplification, you have $ \lnot Q$.

by contrapositive of premmisse $ 1$, you get $ \lnot P.$

Finally, using premisse $ 3$, De Morgan's law and disjunctive syllogism, you have the conclusion $\lnot S$.