I've recently gotten to the ending of Chapter 2 of "Mathematical Proofs: A Transition to Advanced Mathematics", and was doing the supplemental exercises when I came across the following question:
"Let $P(n)$ be an open sentence over the domain $N$ of natural numbers. State the negation of each of the following statements:
a) If $P(n)$ is true for infinitely many $n \in N$, then $P(n)$ cannot be false for infinitely many $n \in N$."
Now, the answer key of the book says that the answer to this would be "If $P(n)$ is true for infinitely many $n \in N$, then $P(n)$ can be false for infinitely many $n \in N$."
However, I ended up with a different answer. Since $P \implies Q$ is logically equivalent to $\neg P \lor Q$, and the negation of $\neg P \lor Q$ is $P \land \neg Q$, I concluded that the negation of the statement in (a) could be written as "$P(n)$ is true for infinitely many $n \in N$, and $P(n)$ is false for infinitely many $n \in N$."
Is this a valid negation for the statement in (a)? I understand that the answer provided by the textbook could be correct, but I'm just curious whether my approach is right or not.