I have to prove that the number of equivalence relations is strictly lower than the number of partial orders over a set A of size >1.
I have managed to prove that the number of different antisymmetric relations over A is strictly higher than the number of symmetric relations over A.
I am struggling trying to connect this information with the reflexive and transitive parts they both share.
Anyway, I do not think that the problem is trying to find explicitly the number, since finding the number of different partial orders for in terms of the size of A seems tonbe a really hard problem.
Any help is appreciated!