# Use Theorem 1.1.1 below to verify the logical equivalence and supply a reason for each step?

Logical Equivalences

I have question about Simplifying Statement Forms, this question

$$\lnot(p \lor \lnot q) \lor(\lnot p \land \lnot q) ≡ \lnot p$$

and this my answer \begin{align} \lnot(p \lor \lnot q) \lor(\lnot p \land \lnot q) &≡ (\lnot p \land \lnot\lnot q) \lor (\lnot p \land \lnot q)&&\text{De Morgan’s laws}\\ &≡(\lnot p \land q) \lor (\lnot p \land \lnot q)&&\text{Double Negative law}\\ &≡p \land (q \lor \lnot q) &&\text{Distributive laws}\\ &≡p \land t &&\text{Negation laws}\\ &≡p &&\text{Identity laws}\\ \end{align}

my answer is correct or not ?

• There is just a typo when you apply the distributive law: you get $\lnot p \land (q \lor \lnot q)$. You keep $\lnot p$ in the next lines. – Taroccoesbrocco Oct 1 '18 at 11:51
• oke thank you're explanation :) – Devo Avidianto P Oct 1 '18 at 11:56

Welcome to MSE! No, its not correct. The third line should be $$\neg p \wedge (q\vee \neg q)$$.