Derivations in sentential logic?

For the first question

~(P → Q)├ P
1.~(~P V Q)
by Material Implication
2.~~P & ~Q
by De Morgan Theorem
3.P & ~Q
by Double Negation
4.P
by Simplification


For the second question

~ ((P & Q) → (P v Q))├ (R & ~S)
1.~ (~(P & Q) V (P v Q))
by Material Implication
2.~~(P & Q) & ~(P v Q)
by De Morgan Theorem
3.(P & Q) & ~(P v Q)
by Double Negation
4.(P & Q) & (~P & ~Q)
by De Morgan Theorem
5.(P & Q)
by Simplification(Using 4)
6.(~P & ~Q) & (P & Q)
by Commutativity(Using 4)
7.(~P & ~Q)
by Simplification(Using 6)
8.P
by Simplification(Using 5)
9 ~P
by Simplification(Using 7)
10. P V (R & ~S)
by Addition (Using 8)
11. (R & ~S)
by Disjunctive Syllogism


For the third question

(((P → Q) → R) → S)├ (P → (Q → (R → S)))
1. ~((P → Q) → R) V S)
by Material Implication
2. ~(~(P → Q) V R) V S)
by Material Implication
3. ~(~(~P V Q) V R) V S)
by Material Implication
4. ~((~~P & ~Q) V R) V S)
by De Morgan Theorem
5. ~((P & ~Q) V R) V S)
by Double Negation
6. (~(P & ~Q) & ~R) V S)
by De Morgan Theorem
7. ((~P V ~~Q) & ~R) V S)
by De Morgan Theorem
8. ((~P V Q) & ~R) V S)
by Double Negation
9. ((~P V Q) V S) & (~R V S)
by Distribution
10.((~P V Q) V S) & (R → S)
by Material Implication
11.(R → S) & ((~P V Q) V S)
by Commutativity
12.(R → S)
by Simplification
13.(R → S) V ~Q
by Addition
14.~Q V (R → S)
by Commutativity
15.Q → (R → S)
by Material Implication
16.(Q → (R → S)) V ~P
by Addition
17.~P V (Q → (R → S))
by Commutativity
18.(P → (Q → (R → S))) "AT LAST @_@"
by Material Implication

for the fourth question
((P & Q) → R)├ (~R → (~Q v ~P))
1. (~R → ~(P & Q))
by Transposition
2. (~R → (~P V ~Q))
by De Morgan Theorem
3. (~R → (~Q V ~P))
by Commutativity