Q:

What are the possible relations on this set?

Let $X=\{x,y\}$ and consider all of the possible relations on $X \times X$
I understand there are 16 possible relations and generally I understand the definitions of reflexive, irreflexive, symmetry, etc... But I'm struggling to write down exactly what the relations here would be.
Please could you give a few examples these relations

A:

The $16$ possible relations are simply all subsets of $\{(x,x),(y,y),(x,y),(y,x)\}$.
Reflexive means that the relation has to include $(x,x)$ and $(y,y)$. There are no other restrictions, so each of $(x,y)$ and $(y,x)$ can either be in or out: $2\times 2$ choices. Irreflexive means that the relation can't include $(x,x)$ or $(y,y)$.
Symmetric means that either $(x,y)$ and $(y,x)$ are both in or both out. In each case there are $4$ options, as again there is no restriction on the other pairs, so there are $8$ in total. 
Asymmetric means "not symmetric", i.e. there is some $a\neq b$ such that $(a,b)$ is in the relation and $(b,a)$ is not, or vice versa. Antisymmetric means that this is true whenever $a\neq b$. For relations on $x,y$ this is the same thing - the only possibility is for $a$ and $b$ to be $x$ and $y$ in some order - but for larger sets they are genuinely different. So there are $8$ asymmetric and $8$ antisymmetric (all the ones which are not symmetric).
For transitive your numbers are wrong again, as there would only be $16$ if all relations on this set were transitive. But $\{(x,y),(y,x)\}$ is not a transitive relation ($x\sim y$ and $y\sim x$ but $x\not\sim x$). In fact there are $13$ transitive relations.

