Represent "No naive is bad" using the first order logic.

Question

How can I represent the following sentence using the first order logic?

"No naive is bad"

I had thought:

$$\neg Naive(x)\vee Bad(x)$$

Answer 1

You can approach this literally, more or less. We are being told that there does not exist an $x$ such that $x$ is both naive ($N(x)$) and bad ($B(x)$).

It makes sense to let "all people" to be the domain (or universe) in which $x$ resides.

$$\begin{align} \lnot \exists x(N(x) \land B(x)) & \equiv \forall x \lnot(N(x) \land B(x)) \\ \\&\equiv \forall x( \lnot N(x) \lor \lnot B(x))\\ \\ &\equiv \forall x (N(x) \rightarrow \lnot B(x))\end{align}$$

Answer 2

Let $Nx$ be "$x$ is naive" and $Bx$ be "$x$ is bad".

Well, if $x$ is naive, $x$ can't be bad. So $$\forall x(Nx\rightarrow \neg Bx)$$