SOLUTION: Stumped C <-> T ~C -> (~S v ~R) ~C -> D (~D v S) & (~D v R) We have to show T

Question 251006: Stumped
C <-> T
~C -> (~S v ~R)
~C -> D
(~D v S) & (~D v R)
We have to show T
Answer by Edwin McCravy(20054) (Show Source): You can put this solution on YOUR website!

 
1.  C <-> T
2.  ~C -> (~S v ~R)
3.  ~C -> D
4.  (~D v S) & (~D v R)
We have to show T
 

5.  ~D v (S&R)                             Distributive law on 4
 
6.  ~C -> ~(S&R)                           DeMorgan's law on the right side of 2 

7.  (S & R) -> C                           Contrapositive of 6 

8.  ~D -> C                                Contrapositive of 3
 
9.  [(S&R) -> C} & (~D -> C)               Conjunction of 7 and 8

10. [~(S&R) v C] & (~~D v C)               Writing conditionals as disjunctions in 9

11. [~(S&R) v C] & (D v C)                 Double negation on D in 10

12. [~(S&R) & D] v C                       Distributive law on 11

13  ~[(S&R v ~D] v C                       DeMorgan's law on the left part of 12

14. (S&R) v ~D                             Commutative law on 5

15. [(S&R) v ~D] & { ~[(S&R) v ~D]  v C }  Conjunction of 14 and 13                          

16.{[(S&R) v ~D] &   ~[(S&R) v ~D]} v C    Associative law on 15

17.    0 v C                               The expression in braces in 16 is a contradiction                      

18.      C                                 The identity law for disjunction in 17

19.      T                                 By biconditional 1


Edwin

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