How to prove that $∃x J(x)$ and $J(m)$ are not logically equivalent?

Question

I'm supposed to use counter models to establish that the two sentences $∃x J(x)$ and $J(m)$ are not equivalent. My initial work is this, does it seem right?

Domain: Lionel Messi, Cristiano Ronaldo

J(x): Plays for Manchester United

m: Lionel Messi

∃x J(x) would be true for Cristiano Ronaldo, but J(m) would be false because Messi does not play for Manchester United. Thus, proving that the sentences are not equivalent logically.

Another question: how big or small should I make my domains when trying to make a counter model? For instance, should it be a list of premier league players (Kevin De Bruyne, Cristiano Ronaldo, etc.) or should it be a broader domain like 'Domain: Premier league players.' Is there an easier and less confusing way?

Answer 1

Domain: Lionel Messi, Cristiano Ronaldo

J(x): Plays for Manchester United

m: Lionel Messi

∃x J(x) would be true for Cristiano Ronaldo

Less confusingly: $∃x J(x)$ is true, not merely for Ronaldo, but for the entire domain including Messi. On the other hand, $J(x)$ is true merely for Ronaldo.

Another question: how big or small should I make my domains when trying to make a counter model? For instance, should it be a list of premier league players (Kevin De Bruyne, Cristiano Ronaldo, etc.) or should it be a broader domain like 'Domain: Premier league players' ?

I don't think this matters. Alternatively, I like Mauro's arithmetical suggestion above, and here's another one: let the domain of discourse be $\{m,7\}$ and define $J(x)$ such that $J(x)$ is true if and only if $x=7.$