Let p and q be propositions. Use Logical equivalences to show that
(p ^ (~ (~ p V q))) V (p ^ q) = p
The idea is to reduce the nesting of parentheses which will then give possibilities of simplification.
Remember to name the rule of simplification as justification.
Start with the left-hand side:
(p ^ (~ (~ p V q))) V (p ^ q)
de Morgan's law:
= (p ^ ( p ^ ~q)) V (p ^ q)
associative properties of ^:
= (p ^ p ^ ~q ) V (p ^ q)
idempotent law (p^p)=p
= ( p ^ ~q ) V (p^q)
distributive law:
= [p V (p^q)] ^ [~q V (p^q)]
distributive law:
= [(pVp)^(pVq)] ^ [(~qVp)^(~qVq)]
associative law:
= (pVp) ^ (pVq) ^ (pV~q) ^ (~qVq)
distributive law:
= (pVp) ^ pV(q^~q) ^ (~qVq)
complement laws:
= (pVp) ^ (pVF) ^ T
identity laws:
= (pVp) ^ p
idempotent law:
= p^p
idempotent law:
= p
QED