Question 188823: I can use: Transportation, Material Implication, Material Equivalence, Exportation, Tautology, Double Negation, Commutation, Association, Distribution, Demorgan's Theorem, Modus Ponems, Modus Tollens, Hypothetical Syllogism, Conjunstion, Simplification, Addition, Constructive Dilemma, Absorption and Disjuntive Syllogism.
10. 1. p -> (q <--> r)
2. ~q -> r
3. ~(q <--> s)
4. p -> s Therefore: ~p
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! I'm sorry, I didn't realize I made a mistake. I'll get this fixed asap...
Sorry for the delay, but here's the corrected solution:
1. p -> (q <--> r)
2. ~q -> r
3. ~(q <--> s)
4. p -> s Therefore: ~p
------------------------
5. ~[(q & s) v (~q & ~s)] 3 Material Equivalence
6. ~(q & s) & ~(~q & ~s) 5 De Morgan's Law
7. ~(~q & ~s) & ~(q & s) 6 Commutation
8. ~(q & s) 6 Simplification
9. ~q v ~s 8 De Morgan's Law
10. q -> ~s 9 Material Implication
11. ~~s -> ~q 10 Transposition
12. s -> ~q 11 Double Negation
13. s -> r 12,2 Hypothetical Syllogism
14. ~(~q & ~s) 7 Simplification
15. ~~q v ~~s 14 De Morgan's Law
16. ~q -> ~~s 15 Material Implication
17. ~q -> s 16 Double Negation
18. ~q -> r 17,13 Hypothetical Syllogism
--------------
19. p -> [(q -> r) & (r -> q)] 1 Material Equivalence
20. ~p v [(~q v r) & (~r v q)] 19 Material Implication
21. [~p v (~q v r)] & [~p v (~r v q)] 20 Distribution
22. [~p v (~r v q)] & [~p v (~q v r)] 21 Commutation
23. ~p v (~r v q) 22 Simplification
24. p -> (~r v q) 23 Material Implication
25. p -> (r -> q) 24 Material Implication
26. (p & r) -> q 25 Exportation
27. (p & r) -> ~s 26,10 Hypothetical Syllogism
28. ~~s -> ~(p & r) 27 Transposition
29. s -> ~(p & r) 28 Double Negation
30. ~q -> ~(p & r) 17,29 Hypothetical Syllogism
31. q v (~p v ~r) 30 Material Implication
32. (q v ~p) v ~r 31 Association
33. ~r v (q v ~p) 32 Commutation
34. ~r v (~p v q) 33 Commutation
35. r -> (p -> q) 34 Material Implication
36. ~q -> (p -> q) 18,35 Hypothetical Syllogism
37. (~q & p) -> q 36 Exportation
38. ~(~q & p) v q 37 Material Implication
39. (~~q v ~p) v q 38 De Morgan's Law
40. (q v ~p) v q 39 Double Negation
41. q v (q v ~p) 40 Commutation
42. (q v q) v ~p 41 Association
43. q v ~p 42 Tautology
44. ~p v q 43 Commutation
45. p -> q 44 Material Implication
46. ~s -> ~p 4 Transposition
47. q -> ~p 10,46 Hypothetical Syllogism
48. p -> ~p 45,47 Hypothetical Syllogism
49. ~p v ~p 48 Material Implication
50. ~p 49 Tautology
|
|
|
