Logic: counting non-isomorphic models in a certain size that satisfy a formula There is a langue L={R}, R is a relation symbol.


*

*How many different models with 2 elements ($|M|=2$) that are not
isomorphic to each other are there? 

*How many different models with 2
elements that satisfy $A=\forall x \forall y (R(x,y) \to (R(y,x))$
are there? (again, that are not isomorphic to each other)


I don't even have a general idea to solve this problem.
From combinatorics, I know there are 16 models. Yet, do I need to check 1 by 1 if they are isomorphic? I assume there is a smarter way to do so. 
 A: The "smart way" to solve this kind of problem uses Burnside's lemma from combinatorics. However, since your problem is so small—there are only $16$ models, and the models have $2$-element universes—the "dumb way" is just as good if not better. You could have figured out the answer in the time you spent typing the question, if you type like I do.
Let's see. A necessary condition for two relations to be isomorphic is that they contain the same number of (ordered) pairs, right? On a two-element universe, the size of the relation can only be $0, 1, 2, 3,$ or $4.$
There is just one relation of size $0$, and one relation of size $4.$
There are $4$ relations of size $1,$ but only $2$ non-isomorphic relations, right? Likewise (taking complements) just $2$ non-isomorphic relations of size $3.$
So the number of non-isomorphic models is $1+2+x+2+1$ where $x$ is the number of non-isomorphic relations of size $2,$ which we haven't counted yet—I leave that to you.
The second question (symmetric relations) is like the first question only easier, because there are fewer models to start with.
