I'm having some trouble with the following question:
Let $\cal L$ be any finite language and let $\cal M$ be a finite $\cal L$-structure. Prove that there is a sentence $\phi$ such that $\cal N\models \phi$ iff $\cal N \cong \cal M$.
I tried the following:
Let $\cal C, F, R$ be the set of constant, functional, and relational symbols in $\cal L$. If $|M|=n$, where $M$ is the universe of $\cal M$, then define $\psi_1$ as follows: $$\psi_1=\exists x_1 ... \exists x_n\forall w \left(\bigvee_{i=1}^n w = x_i \wedge \left(\bigwedge_{i\neq j} x_i\neq x_j\right)\right)$$
Then $\cal N \models \psi_1$ iff $|N| = n$.
Now, define the following equivalence relation in $\cal C$:$$c_i \sim_{\cal M} c_j \iff c_i^{\cal M} = c_j^{\cal M}$$ where $c_i^{\cal M}$ is the interpreation of $c_i$ in the structure $\cal M$. The equivalence classes of $\cal C$ modulo $\sim_{\cal M}$ are $\cal C_1,..., C_k$, which correspond with the sets of constant symbols that in the structure $\cal M$ are interpreted as the same value. Define $$\psi_2=\bigwedge_{i=1}^k\bigwedge_{c_1,c_2\in \cal C_i}c_1 = c_2$$
Then $\cal N \models \psi_1\wedge\psi_2$ iff $|N|=n$ and if $c_i,c_j$ are interpreted as the same in $\cal M$, then they are also interpreted as the same in $\cal N$.
Now, if $\cal F = R = \emptyset$ and $\cal N \models \psi_1\wedge\psi_2$ then we could just choose a bijection $\eta:\cal N\to \cal M$ such that $\eta(c_i^{\cal N})=c_i^{\cal M}$, but I don't know how to proceed when we add functional and relation symbols on top of all of this.
I think, just like what we did with $\cal C$, we need to first account that different symbols in $\cal F, R$ could be interpreted to the same function/relation. I'd start by defining analogously a relation $\sim_{\cal M}$ on each set and work with equivalence relations, but I'm not sure. Am I on the right path here? How should I proceed?