SOLUTION: I need to somehow get "R ≡ ∼Q" as the conclusion, using natural deduction (This means the 18 rules of natural deduction, Modus Ponens, Modus Tullens, De Morgan's Rule e

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Question 928647: I need to somehow get "R ≡ ∼Q" as the conclusion, using natural deduction (This means the 18 rules of natural deduction, Modus Ponens, Modus Tullens, De Morgan's Rule etc etc.)



Conclusion: R ≡ ∼Q
1. A ⊃ (Q ∨ R)
2. (R • Q) ⊃ B
3. A • ∼B
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
 
 1. A->(QvR)
 2. (R&Q)->B	
 3. A&~B
 Conclusion: R<->~Q

 4. ~B               3,simp.
 5. ~B->~(R&Q)       2,CP
 6. ~(R&Q)           5,4,MP
 7. ~Rv~Q            6,DM
 8. ~~R->~Q          7,writing a disjunction as a conditional  
 9. R->~Q            8,DN
10. A                3,simp
11. QvR              1,10,MP
12. ~~QvR            11,DN
13. ~Q->R            10,writing a disjunction as a conditional
14. (R->~Q)&(~Q->R)  9,13,conjunction
15. R<->~Q           14, definition of biconditional   

Edwin