SOLUTION: 1. p <--> q 2. p -> r 3. p -> (r -> s) Therefore q -> s ...so far I've tried this but I got stuck... 4. (p -> q) & (q ->p) 1 equivalence 5. p -> q 4 simplifi

Algebra.Com
Question 188825: 1. p <--> q
2. p -> r
3. p -> (r -> s) Therefore q -> s

...so far I've tried this but I got stuck...
4. (p -> q) & (q ->p) 1 equivalence
5. p -> q 4 simplification
6. (q -> p) & (p -> q)4 Commutation
7. q -> p 6 simplification
8. q -> (r ->s) 7, 3 Hypothetical Syllogism
9. (q & r) -> s 8 exportation
10. (r & q) -> s 9 commutation
11. p -> (p & r) 2 Absorption
please help :(
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
Here's one way to do it... 

1. p <--> q
2. p -> r
3. p -> (r -> s) Therefore: q -> s
------------------------------------
4.  (p -> q) & (q -> p)                      1     Material Equivalence
5.  (q -> p) & (p -> q)                      4     Commutation
6.  q -> p                                   5     Simplification
7.  q -> r                                   6,2   Hypothetical Syllogism
8.  q -> (r -> s)                            6,3   Hypothetical Syllogism
9.  (q & r) -> s                             8     Exportation
10. ~(q & r) v s                             9     Material Implication    
11. (~q v ~r) v s                            10    DeMorgan's Theorem
12. ~q v (~r v s)                            11    Association
13. ~q v (s v ~r)                            12    Commutation
14. (~q v s) v ~r                            13    Association
15. (q -> s) v ~r                            14    Material Implication
16. ~r v (q -> s)                            15    Commutation
17. r -> (q -> s)                            16    Material Implication
18. q -> (q -> s)                            7,17  Hypothetical Syllogism
19. (q & q) -> s                             18    Exportation
20. q -> s                                   19    Tautology

RELATED QUESTIONS

I need help completing this proof. 1. R&S, S->(P->Q), Q->P |- P<->Q (answered by solver91311)
write a direct proof using the eight rules of inference (C → Q) • (~L →... (answered by Edwin McCravy)
Prove the argument: 1. p -> q 2. r \/ s 3. ~s -> ~t 4. ~q \/ s 5. ~s 6. (~p... (answered by Edwin McCravy)
1. ~P∨Q 2. Q→R ∴ P→R 1. R∨~Q 2. P→Q 3. ~R (answered by Edwin McCravy)
I need help finishing this proof. (P<->Q)->R,P->Q,Q->P |-R 1 (1) (P<->Q)->R A 2 (2) (answered by jim_thompson5910)
q+r/q+p (answered by Alan3354)
I saw that people can solve logic proofs on this site. I've looked everywhere for help.... (answered by Edwin McCravy)
P->~(Q&R),Q->R:... (answered by lynnlo)
I cannot solve this: I can use: Transportation, Material Implication, Material... (answered by jim_thompson5910)