# Logical equivalence

1. Sep 8, 2014

### psu12

Hi, I have to prove the following logical equivalence using algebraic substitutions:

(p v ~q) v ~q → (r v p) ∧ ~q ≡ ~r v q → (q v p) ∧ (~p v ~q)

I've already done the truth table for this problem and proved they are logically equivalent but am not sure how to go about using algebraic substitution. The first step I did was changing the if then by using the definition of 'v' but get stuck where to go after that..

2. Sep 12, 2014

### Staff: Mentor

Hi psu12. http://img96.imageshack.us/img96/5725/red5e5etimes5e5e45e5e25.gif [Broken]

You can replace the implication relation on each side, using this equivalence:

x → y ⇔ ~x V y

Then methodically simplify each side using De Morgan's theorems.

Last edited by a moderator: May 6, 2017