Question 188873: I need to solve using 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 <--> q) -> s
2. ~(~r -> t)
3. ~q v ~s Therefore: (t v p) -> (~t & ~q)
AND this is the other problem
1. p v (q & r)
2. ~r
3. p -> (s -> ~t) Therefore: ~(s & t)
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! This is probably too late, but here are the solutions anyway...
# 1
This is one tricky derivation, so you need to be a little creative with this one.
Note: each premise is used and when a new premise is derived from, I separated it to keep things looking clean.
1. (p <--> q) -> s
2. ~(~r -> t)
3. ~q v ~s Therefore: (t v p) -> (~t & ~ q)
--------------------------------------------
4. ~(~~r v t) 2 Material Implication
5. ~(r v t) 4 Double Negation
6. ~r & ~t 5 De Morgan's Law
7. ~t & ~r 6 Commutation
8. ~t 7 Simplification
---
9. ~s v ~q 3 Commutation
10. s -> ~q 9 Material Implication
---
11. (p <--> q) -> ~q 1,10 Hypothetical Syllogism
12. [(p & q) v (~p & ~q) ] -> ~q 11 Material Equivalence
13. ~[(p & q) v (~p & ~q) ] v ~q 12 Material Implication
14. [~(p & q) & ~(~p & ~q) ] v ~q 13 De Morgan's Law
15. ~q v [~(p & q) & ~(~p & ~q) ] 14 Commutation
16. [~q v (~p v ~q)] & [~q v (p v q) ] 15 Distribution
17. ~q v (~q v ~p) 16 Simplification
18. (~q v ~q) v ~p 17 Association
19. ~q v ~p 18 Tautology
20. ~p v ~q 19 Commutation
---
21. ~t & (~p v ~q) 8,20 Conjunction
22. (~t & ~p) v (~t & ~q) 21 Distribution
23. ~(t v p) v (~t & ~q) 22 De Morgan's Law
24. (t v p) -> (~t & ~q) 23 Material Implication
============================================================
# 2
This derivation mainly involves isolating atomic components through modus ponens and a disjunction syllogism.
1. p v (q & r)
2. ~r
3. p -> (s -> ~t) Therefore: ~(s & t)
-----------------------------------------
4. (p v q) & (p v r) 1 Distribution
5. (p v r) & (p v q) 4 Commutation
6. p v r 5 Simplification
7. p 6,2 Disjunctive Syllogism
8. s -> ~t 3,7 Modus Ponens
9. ~s v ~t 8 Material Implication
10. ~(s & t) 9 De Morgan's Law
|
|
|
