Reflexive Relations:

R is a reflexive relation if (a, a) € R,  a € A. It could be noticed if there is at least one member a € A like (a, a) € R, then R is not reflexive.

Symmetric Relations:

R is called a symmetric relation on A if (x, y)€ R →(y, x) € R

That is, y R x when x R y.

It could be noticed that R is symmetric iff R-1 = R

Assume A = {1, 2, 3}, then R = {(1, 1), (1, 3), (3, 1)} is symmetric.

Anti-symmetric Relations:

R is called as a anti-symmetric relation if (a, b) €R  and  (b, a)  €R →a = b

Thus, if a  € b then a can be belongs to b or b can be belongs to a, but never both.

Or, we have never both a R b and b R a apart from when a = b.