We were asked to express some functions using quantifiers then find the negation of the statement, so that no negation is left of a quantifier then express it in simple english.
we were given this statement: "Everyone is rich, but unhappy" and i'm not sure how to express it i thought of these:
∀x(R(x) ∧ F(x)) or ∀x(R(x) → F(x))
On
Assuming $F(x)$ stands for '$x$ is unhappy', you should use
$$\forall x (R(x) \land F(x))$$ because saying that 'everyone is rich but unhappy' is ssaying that everyone is rich and unhappy.
You were told that "the $\land$ doesn;t go with the $\forall$", because often you indeed want to use the $\rightarrow$ with the $\forall$. For example, if we want to say that 'Everyone who is rich is unhappy', we would use $$\forall x (R(x) \rightarrow F(x))$$
Notice that this statement is syaing that everyone who is rich is indeed unhappy, but it does not say that everyone is rich. Indeed, while those who are rich are unhappy, this statement says nothing about those are not rich, because the conditional is trivially true for anyone who is not rich.
You can reword the sentence as "Everyone is unhappy, but rich." Unlike implication, conjunction is a commutative operator.