SOLUTION: INSTRUCTIONS: Use natural deduction to derive the conclusion in the following problems.
Use indirect proof:
1. (R ∨ S) ⊃ (H • ∼G)
2. (K ∨ R) ⊃
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Question 1010728: INSTRUCTIONS: Use natural deduction to derive the conclusion in the following problems.
Use indirect proof:
1. (R ∨ S) ⊃ (H • ∼G)
2. (K ∨ R) ⊃ (G ∨ ∼H)
/ ∼R
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Proof by contradiction
| Number | Statement | Lines Used | Reason |
|---|
| 1 | | (R v S) -> (H & ~G) | | |
| 2 | | (K v R) -> (G v ~H) | | |
| .: | | ~R | | |
| 3 | ~~R | | AIP |
| 4 | R | 3 | DN |
| 5 | R v S | 4 | Add |
| 6 | R v K | 4 | Add |
| 7 | K v R | 6 | Comm |
| 8 | H & ~G | 1,5 | MP |
| 9 | ~G & H | 8 | Comm |
| 10 | H | 8 | Simp |
| 11 | ~G | 9 | Simp |
| 12 | G v ~H | 2,7 | MP |
| 13 | ~H | 12,11 | DS |
| 14 | H & ~H | 10,13 | Conj |
| 15 | | ~(~~R) | 3-14 | IP |
| 16 | | ~R | 15 | DN |
-----------------------------------------------------------------------------
Acronyms/Abbreviations Used
Add = Addition
AIP = Assumption for Indirect Proof
Comm = Commutation
Conj = Conjunction
DM = De Morgan's Law
DN = Double Negation
DS = Disjunctive Syllogism
IP = Indirect Proof (aka proof by contradiction)
MP = Modus Ponens
Simp = Simplification
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