Question 1060204: Using an indirect proof to solve this problem:
1. B ⊃ (C ⊃~B)
2. A ⊃ (B ⊃ C) /~A v ~ B
Thank you!
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! The idea is to assume the complete opposite of the conclusion (statement 3).
Then show how that assumption leads to to a contradiction (statement 14).
This contradiction means that the opposite of the assumption must be true. In other words, the original conclusion is true.
| Number | Statement | Lines Used | Reason |
|---|
| 1 | | B -> (C -> ~B) | | | | 2 | | A -> (B -> C) | | | | :. | | ~A v ~B | | | | 3 | ~(~A v ~B) | | AIP | | 4 | ~~A & ~~B | 3 | DM | | 5 | A & B | 4 | DN | | 6 | B & A | 5 | Comm | | 7 | A | 5 | Simp | | 8 | B | 6 | Simp | | 9 | C -> ~B | 1,8 | MP | | 10 | B -> C | 2,7 | MP | | 11 | B -> ~B | 10,9 | HS | | 12 | ~B v ~B | 11 | MI | | 13 | ~B | 12 | Taut | | 14 | B & ~B | 8,13 | Conj | | 15 | | ~A v ~B | 3-14 | IP |
Abbreviations/Acronyms Used
AIP = Assumption for Indirect Proof
Comm = Commutation
Conj = Conjunction
DM = De Morgan's Law
DN = Double Negation
HS = Hypothetical Syllogism
IP = Indirect Proof
MI = Material Implication
MP = Modus Ponens
Simp = Simplification
Taut = Tautology
|
|
|
