Finding number of relations on a set with 3 elements

Question

How do I find find out how many non reflexive relations X on the set P = {1, 2, 3}?

I know $2^{n^2 - n}$ returns how many reflexive relations there on a set. Do I subtract that from something to get my result?

Answer 1

A relation on set $P$ is a subset of $P\times P$. In this case, $|P|=3$ so $|P\times P|=9$. Hence there are $2^9$ subsets of $P\times P$, and thus $2^9$ relations on $P$.

Answer 2

Non-Reflexive relations are simply those that do not satisfy the aRa criteria for all 'a' in the corresponding set. Therefore the number of these is mostly irrelevant.

Now, Irreflexive relations are those relations such that for every a in the set, aRa is False. They are in stark contrast to reflexive relations.

Therefore, the number of irreflexive relations on a set of cardinality n is easily the difference of the total number of binary relations possible on the set and the total number of reflexive relations on the set.

=2^(n^2)-2^(n(n-1))