A relation R defined on a non-empty set A is called anti-symmetric if

NOTE- 

Whether the empty relation is reflexive or not depends on the set on which you are defining this relation -- you can define the empty relation on any set XX.

1- The statement "RR is reflexive" says: for each x∈Xx∈X, we have (x,x)∈R(x,x)∈R. This is vacuously true if X=∅X=∅, and it is false if XX is nonempty.

2- The statement "RR is symmetric" says: if (x,y)∈R(x,y)∈R then (y,x)∈R(y,x)∈R. This is vacuously true, since (x,y)∉R(x,y)∉R for all x,y∈Xx,y∈X.

3- The statement "RR is transitive" says: if (x,y)∈R(x,y)∈R and (y,z)∈R(y,z)∈R then (x,z)∈R(x,z)∈R. Similarly to the above, this is vacuously true.

4- To summarize, RR is an equivalence relation if and only if it is defined on the empty set. It fails to be reflexive if it is defined on a nonempty set.