Hello.

If I for example have a set $\displaystyle A=\{1,2,3\}$, I know that a binary relation on the set $\displaystyle A$ is any subset of $\displaystyle A \times A=\{(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)\}$. (I hope I'm correct so far!) The problem however is how many different binary relations there is?