I know there is a lot of information on this internet for this, I've been going through it the past 30 minutes. I'm getting confused to if the answer is actually 203 relations, because when I try to calculate only reflexive, symmetry, and both reflexive & symmetry I get much higher numbers then 203.
Help is appreciated.
A relation is any subset of $A \times A$. Given any set $X$, the number of subsets is $2^{\lvert X \rvert}$.
So the number of relations on a 6-element set is $2^{36}$. Big number! For example one relation could be $\{(1,2)\}$. In other words 1R2 and that's the entire relation. You can see that there are a lot of possible relations.
An equivalence relation is the same as a partition, so the Bell numbers tell you how many equivalence relations there are on a given set. But you are asked to find how many relations there are of any type. So the partitions aren't relevant here.