This is a past paper question for my revision. L is defined as first order language with only the unary function {f} as its non-logical symbol.
I have so far found a sentence for the first part of c - it uses a function symbol, and I believe that for the second part of c I cannot find such a sentence without a predicate or function symbol, though I can't figure out how to prove this. If so, then in part d the set of sentences can't be finite, else we could take the conjunction of them as our sentence in this part of c.
I am also struggling to find a set of sentences as asked for in part d. I can't really think of how to split A into two parts without a function or predicate symbol.
This answer addresses part d. Define sentences $\mu_n$ and $\nu_n$ for $n = 1, 2, \ldots$ as follows: $$ \begin{align*} \mu_n &\equiv \exists x_1\exists x_2 \ldots \exists x_n\forall x(x = x_1 \lor x = x_2 \lor \ldots x = x_n) \tag*{$n = 1, 2, \ldots$}\\ \nu_1 &\equiv \mu_1 \\ \nu_n &\equiv \mu_n \land \lnot \mu_{n-1} \tag*{$n = 2, 3, \ldots$} \end{align*} $$ So ${\cal A} \models \mu_n$ asserts that $\cal A$ has at most $n$ elements and ${\cal A} \models \nu_n$ asserts that it has exactly $n$ elements. Now let $\Psi = \{\lnot \nu_1, \lnot \nu_3, \ldots, \nu_{2n+1}, \ldots\}$. Then if $\cal A$ is a finite structure, ${\cal A} \models \Psi$ iff $\cal A$ has an even number of elements.