# Thread: Tautology problem

1. ## Tautology problem

So I am currently working on a problem that asks me to prove

[(p -> q) ^ (q -> r)] -> (p -> r) is a tautology by using logical equivalences.

I just enrolled in the course, so I don't know if this is by any means correct, but here is what I have come up with. Any assistance would be great.

[(~p v q) ^ (~q v r)] -> (p -> r) by implication

[(~p v q) ^ (~r v q)] -> (p -> r) contrapositive

(~p v q) -> (p -> r) transitivity (idk if there is another term?)

(p -> q) -> (p -> r) implication

(q -> r) transitivity

2. Hi, giaky

I'm not sure that I understand you correctly... You're not sure in your proof?
(~p v q) -> (p -> r) transitivity (idk if there is another term?)
I'm not sure in correctness of this operation

Here is my proof. :

[(p -> q) ^ (q -> r)] -> (p -> r)

[(~p v q) ^ (~q v r)] -> (~p v r) ---- by implication

~[(~p v q) ^ (~q v r)] v (~p v r) ---- by implication

[~(~pvq)v~(~qvr)]v(~pvr) --- De Morgan's law

[p^~q)v(q^~r)v(~pvr) --- De Morgan's law

[p^q = pq ----- notation]
[~p='p ----- notation]

pq v qr v p v r ====

now we'll use Distributive Property: x+yz = (x+y) (x+z)
and Basic Identity: x+x’ = 1

pq v p = (p v p)(q v p)=q v p
qr v r = (q v r)(r v r) = q v r

=== q v p v q v r = `p v r = p->r

p.s. sorry if there is misunderstanding...I'm just trying to help