Question 188928: I cannot solve this: 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.
1. ~(p <--> s)
2. (p v q) -> r
3. (t v ~s) -> (~s & ~r) Therefore: ~t & ~p
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! This is a very long proof. I hope that I'm not too late. There are a number of ways to derive this, but the main goal is isolate ~t and ~p (see lines 41 and 44) and then form a conjunction of the two atomic components.
Note: when I start deriving from new premise, I separate the lines (to make things a little bit cleaner). Normally those dividing lines wouldn't be in the proof.
1. ~(p <--> s)
2. (p v q) -> r
3. (t v ~s) -> (~s & ~r) Therefore: ~t & ~p
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4. ~[(p & s) v (~p & ~s)] 1 Material Equivalence
5. ~(p & s) & ~(~p & ~s) 4 De Morgan's Law
6. ~(~p & ~s) & ~(p & s) 5 Commutation
7. ~(p & s) 5 Simplification
8. ~p v ~s 7 De Morgan's Law
9. ~s v ~p 8 Commutation
10. s -> ~p 9 Material Implication
11. ~(~p & ~s) 6 Simplification
12. ~~p v ~~s 11 De Morgan's Law
13. ~~p v s 12 Double Negation
14. ~p -> s 13 Material Implication
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15. ~(p v q) v r 2 Material Implication
16. (~p & ~q) v r 15 De Morgan's Law
17. r v (~p & ~q) 16 Commutation
18. (r v ~p) & (r v ~q) 17 Distribution
19. r v ~p 18 Simplification
20. ~p v r 19 Commutation
21. p -> r 20 Material Implication
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22. ~(t v ~s) v (~s & ~r) 3 Material Implication
23. (~t & ~~s) v (~s & ~r) 22 De Morgan's Law
24. (~t & s) v (~s & ~r) 23 Double Negation
25. [(~t & s) v ~s] & [(~t & s) v ~r] 24 Distribution
26. [(~t & s) v ~r] & [(~t & s) v ~s] 25 Commutation
27. (~t & s) v ~s 25 Simplification
28. ~s v (~t & s) 27 Commutation
29. s -> (~t & s) 28 Material Implication
30. (~t & s) v ~r 26 Simplification
31. ~r v (~t & s) 30 Commutation
32. r -> (~t & s) 31 Material Implication
33. p -> (~t & s) 21,32 Hypothetical Syllogism
34. ~p v (~t & s) 33 Material Implication
35. (~p v ~t) & (~p v s) 34 Distribution
36. (~p v s) & (~p v ~t) 35 Commutation
37. ~p v s 35 Simplification
38. p -> s 37 Material Implication
39. p -> ~p 38,10 Hypothetical Syllogism
40. ~p v ~p 39 Material Implication
41. ~p 40 Tautology
42. s 14,41 Modus Ponens
43. ~t & s 29,42 Modus Ponens
44. ~t 43 Simplification
45 ~t & ~p 44,41 Conjunction
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