Basic Predicate Logic Translations

let T be a set of all teams , let G be a set of all groups , Apartof “team t is in group g,” where t ∈ T and g ∈ G Japan(t) “team t is in Japan,” where t ∈ T FIFA(t) " team t played in the FIFA final" where t ∈ T

how would you translate Every team is in exactly one group.

Is this an answer or is it way off?

∀g∈G,∃t∈T, Apartof(t, g)∧∃z∈T, z =!(not equal) t

what would be someother ways of translating this to logic or is there only one way?

$\forall t \in T \exists g \in G (ApartOf(t,g) \land \forall z \in G (ApartOf(t,z) \rightarrow z = g))$
$\forall t \in T \exists g \in G (ApartOf(t,z) \land \neg \exists z\in G (ApartOf(t,z) \land z \not = g))$