# Thread: Showing a compuond proposition is a tautology using logical equivalences

1. ## Showing a compuond proposition is a tautology using logical equivalences

Hi all,

I am currently working on showing that [p AND (p IMPLIES q)] IMPLIES q is a tautology and, really just don't know where to begin. Not expecting any answers, but to be pointed in the correct direction.

2. ## Re: Showing a compuond proposition is a tautology using logical equivalences

My first choice would be a truth table but if you must use "logical equivalences" you might start with "A implies B" is equivalent to "not B implies not A". Here that would be "not q implies not (p and (p implies q))"

3. ## Re: Showing a compuond proposition is a tautology using logical equivalences

Thank you. My first choice would have been a truth table but the professor said no specifically to the truth table.

4. ## Re: Showing a compuond proposition is a tautology using logical equivalences

Hello, michaelgg13!

Darn! Latex isn't working . . .

Show that: .[p ∧ (p → q)] → q

I'll list the steps.
I'll let you supply the reasons.

We have:

[p ∧ (~p ∨ q)] → q

[(p ∧ ~p) ∨ (p ∧ q)] → q

[F ∨ (p ∧ q)] → q

(p ∧ q) → q

~(p ∧ q) ∨ q

(~p ∨ ~q) ∨ q

~p ∨ (~q ∨ q)

~p ∨ T

. . .T