Let A = {1, 2, 3, . . . , n}, and let R be a relation on A that is antisymmetric.
(a) What is the maximum number of ordered pairs that can be in R?
(b) How many antisymmetric relations on A have the size you found in (a)?
Do you know how to make a table of ordered pairs?
$\displaystyle \begin{array}{lll}
{(a,a)} & {(a,b)} & {(a,c)} \\
{(b,a)} & {(b,b)} & {(b,c)} \\
{(c,a)} & {(c,b)} & {(c,c)} \\ \end{array} $
That is on a set of three elements. There are $\displaystyle 9$ pairs.
So there are $\displaystyle 2^9$ possible relations on a set of three.
When is a relation not antisymmetric?
There is a start.
If you are unable to carry on, then you need to go to instructor for face-to-face help.