If we have
A(X) = x is in this arcade
W(X) = "x won a grand prize"
P(x) = "x played Pac-Man"
how would I express "A person in this arcade won a grand prize," and "Nobody in this arcade played Pac-Man." As premises in a quantified logical statement.
What I have done so far is $\forall x(\neg A(x) \implies P(x))$ and $\exists x (A(x) \implies W(x)$