Hello mikeqwertyThe definition simply means that the only way you can haveaRbandbRaat the same time in an antisymmetric relation is ifaandbare one and the same element. Don't confuse this with a reflexive relation which says thataRaforallelementsa.

There are some pretty straightforward examples here: Antisymmetric relation - Wikipedia, the free encyclopedia. If you don't understand them, tell us where you're having trouble, and we'll take it from there.

As far as the digraph of an antisymmetric relation is concerned, ifaandbare two distinct points (nodes) on the digraph, and there's a directed link fromatob, then there can't be a link back frombtoa. If there were a link back, then you would have a pair of elementsaandbfor whichaRbandbRawithoutabeing equal tob.

Grandad

PS The answer to your second post: No.