SOLUTION: I’m doing homework concerning the rules of replacement. We are allowed to use 8 rules of implication (modus ponens, modus tollens, pure hypothetical syllogism, disjunctive syllogis
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Question 1102118: I’m doing homework concerning the rules of replacement. We are allowed to use 8 rules of implication (modus ponens, modus tollens, pure hypothetical syllogism, disjunctive syllogism, constructive dilemma, simplification, conjunction, and addition) and 10 rules of replacement (De Morgan’s rule, commutativity, associativity, distribution, double negation, transposition, material implication, material equivalence, exportation, and tautology). I have been stuck on the same proof for several days, trying different solutions for multiple hours each day. The problem is as follows:
Premise 1: If Q, then (A and V)
Premise 2: If (If not V, then C), then B
Conclusion: If Q, then B
Even a nudge in the right direction would be appreciated. Thank you.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Notes about notation:
The tilde indicates "not". Example: ~A = not A
The arrow "->" without quotes is the implication "if..then..". For example writing "A -> B" means "if A, then B"
The ampersand & represents conjunction. Example: A & B = A and B
The vertical bar | represents disjunction "or". Example: A | B = A or B. I would use a lowercase 'v' for the disjunction operator, but I'm using a vertical bar to avoid confusion with the V already there
Here is one way to do the derivation:
| Number | Statement | Lines Used | Reason |
|---|
| 1 | Q -> (A & V) | | |
| 2 | (~V -> C) -> B | | |
| :. | Q -> B | | |
| 3 | ~Q | (A & V) | 1 | Material Implication |
| 4 | (~Q | A) & (~Q | V) | 3 | Distribution |
| 5 | (Q -> A) & (Q -> V) | 4 | Material Implication |
| 6 | (Q -> V) & (Q -> A) | 5 | Commutation |
| 7 | Q -> V | 6 | Simplification |
| 8 | ~(~V -> C) | B | 2 | Material Implication |
| 9 | ~(V | C) | B | 8 | Material Implication |
| 10 | (~V & ~C) | B | 9 | De Morgan's Law |
| 11 | B | (~V & ~C) | 10 | Commutation |
| 12 | (B | ~V) & (B | ~C) | 11 | Distribution |
| 13 | B | ~V | 12 | Simplification |
| 14 | ~V | B | 13 | Commutation |
| 15 | V -> B | 14 | Material Implication |
| 16 | Q -> B | 7,15 | Hypothetical Syllogism |
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