It can be reflexive, but it can't be symmetric for two distinct elements. Asymmetric is the same except it also can't be reflexive. An asymmetric relation never has both aRb and bRa, even if a = b. So an asymmetric relation is just one that is both antisymmetric and irreflexive.

Can a relation be irreflexive and antisymmetric?

No, an antisymmetric relation doesn't have to be irreflexive. However, if a relation R is both transitive and irreflexive, then it is automatically antisymmetric.

Are symmetric relations transitive?

R is reflexive if for all x A, xRx. R is symmetric if for all x,y A, if xRy, then yRx. R is transitive if for all x,y, z A, if xRy and yRz, then xRz.