Translation of natural language to relational proposition Here are some preconditions:

*

*unary relation constant: person, celebrity, governor

*binary relation constant: knows$(x,y)$ means $x$ knows $y,$ differ$(x,y)$ means $x$ is different to $y$
I have 2 translation questions; are my answers correct?
i)  There is no celebrity or a person that knows any governor.
$$ \text{∀x.∃y.((celebrity(x) ∨ person(x) ∧ governor(y)) → ¬knows(x, y)}$$
ii) Every person knows three celebrities.
$$  \text{∀w.∃x.∃y.∃z.(person(w) ∧ celebrity(x) ∧ celebrity(y) ∧ celebrity(z)}\\\text{∧ knows(w, x) ∧ knows(w, y) ∧ knows(w, z))}$$
 A: *

*There is no celebrity or a person that knows any governor.
$$∀x.∃y.((\text{celebrity}(x) ∨ \text{person}(x) \land
\text{governor}(y)) → ¬\text{knows}(x, y).$$
Correction:
$$∀x ∀y\;\bigg((\big(\text{celebrity}(x) ∨ \text{person}(x)\big)
\land \text{governor}(y)) → ¬\text{knows}(x, y)\bigg).$$


*Every person knows three celebrities.
$$∀w.∃x.∃y.∃z.(\text{person}(w) \land \text{celebrity}(x) \land
\text{celebrity}(y) \land \text{celebrity}(z) \\\land \text{knows}(w,
x) \land \text{knows}(w, y) \land \text{knows}(w, z)).$$
Your translation is actually equivalent to $$\;∀w \;\text{person}(w) \land  ∀w ∃x\;\bigg(\text{celebrity}(x) \land \text{knows}(w, x)\bigg),$$ i.e., "Everything is a person, each of whom knows some celebrity."
Correction: $$∀p ∃x ∃y ∃z ∀c\;(\text{person}(p)\to \bigg(x\neq y\land y\neq z\land x\neq z  \\\land (\big(c=x\lor c=y\lor c=z\big) \to\big(\text{celebrity}(c)\land\text{knows}(p, c)\big) )\bigg)).$$
P.S. For #2, if the statement becomes "every person knows exactly three celebrities", then just change the final $\to$ to $\leftrightarrow.$
