Discrete Math question, Logical connectives Associative law example

Question

My book says that $\lnot(P ∨ ((\lnot P) ∧ Q)) ≡ ((\lnot P) ∧ P) ∨ ((\lnot P) ∨ (\lnot Q))$

However, when I follow the Distributive law for $(\lnot P) ∧ (P ∨ (\lnot Q))$ I get

$((\lnot P) ∧ P) ∨ ((\lnot P) ∧ (\lnot Q)) ≢ ((\lnot P) ∧ P) ∨ ((\lnot P) ∨ (\lnot Q))$

Is the book wrong? Or am I missing something?

The book is called Discrete Mathematics by Gary Chartrand and Ping Zhang 2011

Question is #21 in Supplementary Exercises for Chapter 1

Answer 1

NO; the book has:

$\lnot [P \lor (\lnot P \land Q)] \equiv (\lnot P \land \lnot Q)$.

First apply Distributivity:

$\lnot[P \lor (\lnot P \land Q)] \equiv \lnot[(P \lor \lnot P) \land (P \lor Q)]$.

Then De Morgan.