I just have started to study some model theory and I have difficulties with following two problems:
Prove that every theory of signature, consisting only of a finite number of unary predicate symbols and equality, has an at most countable number of countable models up to isomorphism.
And
Prove that every theory of signature, consisitng only of binary predicate $R$ and equality, with axioms $$\forall x \forall y (R(x,y) \rightarrow R(y,x))$$ $$\forall x R(x,x)$$ has exactly continuum countable models up to isomorphism.
For the first problem, I came up with the following. Instead of models of theories let's investigate models of signature. Fix the order of predicate symbols. Then we can introduce function $ \phi: M \rightarrow \{0,1\}^n$ which assigns to each element of the model n-tuple of values of predicates on this element.
Now we also can introduce an equivalence relation on $M$: two elements are equivalent if they are assigned equal n-tuples. Two models are isomorphic iff corresponding equivalence classes have the same cardinality. If we have $n$ predicate symbols, then there are $2^n$ tuples, which define a partition of classes. Let's choose an arbitrary model. We can define a function $\psi : M \rightarrow \{1, 2, 3, ..., 2^n\}$ which also determine partition. There are countably many such functions. Therefore, there are at most countable number of models up to isomorphism.
For the second problem, I don't have any clear ideas.
Any hints and ideas are appreciated.
Thanks!