What is the difference between a “reflexive” and an “anti-symmetric relation”?

Answer 1

A reflexive relation relates everything to itself. One such relation would be a relation on the real numbers R(x, y): x = y, as for any real number a, R(a, a) is valid. (because a = a for any given number.)

A symmetric relation means that for any case where R(a, b), also R(b, a).

Antisymmetric relations are almost the exact opposite- Imagining the relation as a directed graph, none of the arrows point both ways (except when something points to itself.)

Mathematically, when R(a, b), if R(b, a), a = b.

Or, when a =/= b, if R(a, b), not R(b, a).

Note that an "empty relation" which doesn't relate anything to anything else is both symmetric and antisymmetric (but not reflexive!).