Which of the following formalizes the definition of the predicate \(Pr(x)\) over the set of integers, and evaluates to \(T\) exactly when \(x\) is prime. (Select all and only correct options.)
\(\forall a \in \mathbb{Z}^{\neq 0}~( ~(x > 1 \land a >0) \to F(~(a,x)~))\)
\(\lnot \exists a \in \mathbb{Z}^{\neq 0} ~(x > 1 \land (a=1 \lor a=x) \land F(~(a,x)~))\)
\((x > 1) \land \forall a \in \mathbb{Z}^{\neq 0}~( ~(~ a>0 \land F(~(a,x)~)~) \to (a=1 \lor a=x)~)\)
\((x > 1) \land \forall a \in \mathbb{Z}^{\neq 0}~( ~(~ a>1 \land \lnot (a=x) ~) \to \lnot F(~(a,x)~)~)\)