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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?

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Since the sentence is about every team, you need to universally quantify over the teams. Moreover, since each team will be in one group, you will need an existential for that. So, it is just the other way around:

$\forall t \in T \exists g \in G (ApartOf(t,g) \land \forall z \in G (ApartOf(t,z) \rightarrow z = g))$

Alternatively:

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

... And there are other translations possible as well ... In fact, you can in theory come up with infinitely many sentences that are equivalent ... But only some are actually pretty 'direct' translations.

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