How to solve Logical Equivalence

Question

okay I have an equivalence that I have to prove. these equivalences use biconditionals and boolean algebra. but I'm not sure which order I apply the laws of logical equivalence.

EDIT:

It is an assignment. I've been stuck on it for days getting different answers different times.

-(p V (q ^ -r)) ^ q == (-p ^ q) ^ r

I don't necessarily want anyone to solve it, I want them to explain, in a generic way, how they would go about solving it and the order they apply the laws. that way I can learn from it.

Answer 1

Well there's only so many equivalencies to go around. Through these kinds of classes and mathematics ones too, commonly you didn't have but a few tricks in the toolbox so to speak. De Morgans laws, etc. A chapter might mention like 4-5 rules and you should definitely try all of those.

Might help to work from the opposite side too if you've only been working the equation in one direction. It's very much a play around with it until you get enough experience to identify a better initial guess. But as you play with it you'll start to see a few manipulations to get reasonably close and then the problem solving comes in to figure out the 1-2 manipulations left you aren't seeing. Even my best professors told me you have to beat your head on a desk til it works.

Answer 2

-(p V (q ^ -r)) ^ q

-((p V q) ^ (p V -r)) ^ q [by distr of V over ^]

(-(p V q) V -(p V -r)) ^ q [by DeMorgan]

((-p ^ -q) V (-p ^ r)) ^ q [by DeMorgan]

(-p ^ -q) ^ q V (-p ^ r) ^ q [by distr of V over ^]

-p ^ (-q ^ q) V (-p ^ r) ^ q [by assoc of ^]

-p ^ F V (-p ^ r) ^ q [by A ^ -A == F, where F is falsity (0)]

F V (-p ^ r) ^ q [by A ^ F == F, where F is falsity (0)]

(-p ^ q) ^ r [by A v F == A, where F is falsity (0)]