Translate the following argument into symbolic notation.
Use $P(x)$ for "$x$ is purple",
$Q(x,y)$ for "$x$ questions $y$" and
$R(x)$ for "$x$ is ridiculous".
Somebody who is purple questions somebody.
Somebody who is not purple questions somebody.
Everybody who questions anybody is ridiculous.
Therefore, at least two people are ridiculous.
I have so far done the following; however, I am not sure if I am correct. Once I find the notation to the sentences I have to find a proof which I understand how to. But before that I need to make sure I have the symbolic notation correct. Can someone please help check where I it’s wrong, or a hint will do as well.
$\exists x (P(x) \wedge Q(x,y))$
$\exists x (\neg P(x) \wedge Q(x,y))$
$\forall x \bigl(P(x)\implies \exists y(R(x))\ \bigr)$
Therefore, $\exists x (R(x))$