Question 1198762: Use Indirect proof to solve the following:
(P v F) ⊃ (A v D)
A ⊃ (M • ~P)
D ⊃ (C • ~P) / ~P
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
This is one way to do the derivation.
| Number | Statement | Line(s) Used | Reason | | 1 | | (P v F) ⊃ (A v D) | | | | 2 | | A ⊃ (M • ~P) | | | | 3 | | D ⊃ (C • ~P) | | | | :. | ~P | | | | 4 | ~(~P) | | Assumption For Indirect Proof | | 5 | P | 4 | Double Negation | | 6 | P v F | 5 | Addition | | 7 | A v D | 1,6 | Modus Ponens | | 8 | (M • ~P) v (C • ~P) | 2,3,7 | Constructive Dilemma | | 9 | (M v C) • ~P | 8 | Distribution | | 10 | ~P | 9 | Simplification | | 11 | ~P • P | 10,5 | Conjunction | | 12 | | ~P | 4 - 11 | Indirect Proof |
The idea is to start with the conclusion ~P and negate it to get ~(~P).
The goal is to show a contradiction arises when we assume the opposite of the conclusion.
As shown above, the contradiction happens on line 11 when we have ~P and P together.
This contradiction then leads us to conclude the opposite of the assumption ~(~P) must be the case, i.e. the original conclusion we started with is the case.
This fully wraps up the proof.
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