The statements are:
- $\forall x \exists y(fx = y)$
- $\forall x \exists x(fx = x)$
- $\exists x(fx = x)$
I know that two statements $\alpha, \beta$ are equivalent iff $I \vDash (\alpha)\equiv (\beta)$ for every interpretation $I.$
It might be wise trying to prove $I \vDash (\alpha) \rightarrow (\beta)$ and $I \vDash (\alpha) \leftarrow (\beta)$ seperately.
But how do I do that?
Obviously, the first statement doesn't imply the second, whereas the second statement does indeed imply the first. The second statement describes a constant function, whereas the first statement describes a function in general. Of course every constant function is a function, but not the other way around. But I don't think that this is a proper answer.