While studying about binary relations, I got confused while solving some simple problems. For example: Let $R=\{(a,a), (a,c), (b,b), (b,c), (c,c), (c,a), (c,b), (d,d)\}$ be a binary relation on the set $A=\{a, b, c, d\}$
Are $\{(a,c), (c,a)\}$ and $\{(b,c), (c,b)\}$ enough to meet the criterion of symmetry or all combinations of $a$, $b$, $c$ and $d$ must be present for that? Respectively, is $\{(a,c), (c,b)\}$ enough to meet the criterion of transitivity?
The laws involved in defining binary relations are universal laws. It means that you have to verify them for ALL the elements on which the binary relation $R$ is defined.
For example regarding symmetry, for any couple $(x,y) \in R$, you have to verify that $(y,x)$ is also belonging to $R$. This is the case in your example. Apply the same verifications for reflexivity and transitivity.