2. Let varphi be the following sentence in a first-order language with one one-place function symbol f: \varphi=((\forall x \exists y(x=f y)) \wedge(\forall x \forall y(f x=f y \rightarrow x=y))) (i) Give a short explanation of what it means for a structure to satisfy the sentence varphi above. (ii) Show that there are types of countable structures which satisfy p.uncountably many different isomorphism (iii) For each n > 0, let psin be the sentence \forall x(\neg(x=\underbrace{f \cdots f}_{n \text { times }} x)) Show that there are only countably many different isomorphism types of countable structures which satisfy both varphi and psin for all n > 0.

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