SOLUTION: Given: ~C or D (~A or B) -> F ~B -> C ~F Prove: D

Question 178748: Given: ~C or D
(~A or B) -> F
~B -> C
~F
Prove: D
Answer by Edwin McCravy(20056) (Show Source): You can put this solution on YOUR website!



(1)  ~C \/ D 
(2)  (~A \/ B) -> F 
(3)  ~B -> C 
(4)  ~F 
Prove: D

   I.  ~F                          Given 

  II.  ~(~A/\~B)                   By ~F -> ~(~A\/B), the contrapositive of (2)

 III.  ~(~A)/\~B                   ~(~A\/B) <-> ~(~A)/\~B DeMorgan's law

  IV.  A/\~B                       ~(~A)<-> A  Negation of negation 

   V.  ~B                          Conjunction IV                   

  VI.  C                           V, and (3),  ~B -> C is given

 VII.  ~C \/ D                     given

VIII.  C /\ (~C \/ D)              Conjunction of VI with VII
 
  IX.  (C /\ ~C) \/ (C /\ D)       Distributive law with VIII

   X.  false \/ (C /\ D)           Conjunction of C with its negation is false 

  XI.  C /\ D                     "false" is the identity for disjunction       

 XII.  D                           Conjunction XI

Edwin

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