question about laws of logical equivalence

Question

i've this question which I have to solve using laws of logical equivalence but I can't. I've been trying to solve this since a few hours now.

p ⊕ ( ¬ p ∧ q) ≡ p ∨ q

I tried to solve the LHS using a ⊕ b = (a V b) ∧ ¬ (a ∧ b) but can't complete the question as I keep getting stuck. what i've tried so far on the LHS

[p V (¬p ∧ q)] ∧ ¬[p ∧ (¬p ∧ q)] (using the expression above)

[(p V ¬p) ∧ (p V q)] ∧ ¬[p ∧ (¬p ∧ q)] (using [a V (b ∧ c)] ≡ [(a V b) ∧ (a V c)])

[T ∧ (p V q)] ∧ ¬[p ∧ (¬p ∧ q)] (using (a V ¬a) ≡ T)

(p V q) ∧ ¬[p ∧ (¬p ∧ q)] (using (T ∧ a) ≡ a)

I can't go further, any help would be appreciated.

Answer 1

Hint

You have started correctly unwinding exclusive or :

$[p \lor (\lnot p \land q)] \land \lnot [p \land (\lnot p \land q)]$.

The RHS is simply $\text T$, because $(p \land \lnot p) \equiv \text F$.

Then, we have to consider that $\alpha \land \text T \equiv \alpha$; thus, the formula amounts to the LHS only.

And you have already simplified it to $(p \lor q)$.