Question

Let p and q be propositions. Use Logical equivalences to show that

(p ^ (~ (~ p V q))) V (p ^ q) = p

Answer 1

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