SOLUTION: Note that ‘->’ is used for conditionals, ‘~’ is used for negations, ‘v’ is used for disjunctions, ‘&’ is used for conjunctions, ‘<->’ is used for biconditionals, and ‘/’ is used as

Algebra ->  Proofs -> SOLUTION: Note that ‘->’ is used for conditionals, ‘~’ is used for negations, ‘v’ is used for disjunctions, ‘&’ is used for conjunctions, ‘<->’ is used for biconditionals, and ‘/’ is used as      Log On


   

Question 1041047: Note that ‘->’ is used for conditionals, ‘~’ is used for negations, ‘v’ is used for disjunctions, ‘&’ is used for conjunctions, ‘<->’ is used for biconditionals, and ‘/’ is used as the conclusion indicator.
Prove the following three arguments to be valid using the method of Natural Deduction. Do not use Conditional Proof in these proofs.
1. ~(P v Q)
2. (R v S) -> P / ~S

1. P v (Q v R)
2. (~T & ~R) / (Q v P)

1. (P v Q) -> R
2. (S v R) -> T
3. ~(Q -> E) / T
Answer by robertb(5830) About Me  (Show Source):
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1. ~(P v Q)
2. (R v S) -> P / ~S
3. ~P&~Q ............De Morgan's
4. ~P .............Simplification
5. ~(R v S) v P .....Material implication on 2
6. ~(R v S) v ~~P ....Double negation
7. ~(R v S) .........Disjunctive syllogism on 4 and 6
8. ~R&~S ...........De Morgan's on 7
9. ~S ...............Sinplification
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1. P v (Q v R)
2. (~T & ~R) / (Q v P)
3. ~R .............Simplification on 2.
4. ( P v Q) v R .... Associativity on 1
5. P v Q ..........Disjunctive syllogism on 3 and 4
6. Q v P ............Commutativity
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1. (P v Q) -> R
2. (S v R) -> T
3. ~(Q -> E) / T
4. ~(P v Q) v R .......MI on 1
5. (~P & ~Q) v R ......Dl on 4
6. (~P v R) & (~Q v R) ...Distributivity on 5
7. ~(S v R) v T .......MI on 2
8. (~S & ~R) v T ......Dl on 7
9. (~S v T) & (~R v T) ...Distributivity on 5
10. ~Q v R .............simplification on 6
11. Q -> R ............MI on 10
12. ~R v T .............simplification on 9
13. R -> T ............MI on 12
14. Q -> T ............hypothetical syllogism on 11 and 13
15. ~Q v T ............MI on 14.
16. ~(~Q v E) .........MI on 3.
17.~~Q & ~E ............distribution
18. Q & ~E ............double negation
19. Q .................simplification on 18
20. ~~Q .............DN
21. T ...............disjunctive syllogism on 15 and 20.