Def:

R is irreflexive iff.

Not irreflexive iff.

R is asymmetric iff. if , then

Not asymmetric iff. and

R is anti-symmetric iff. if , then .

Not anti-symmetric iff.

Proofs:

R is symmetric implies that is symmetric.

Let in R. Therefore, . By the symmetric hypothesis, . Thus, we have . We now have a path of length from b to x to a in R and is symmetric.

If R is asymmetric, then

By contraction: Suppose R is asymmetric and

Let . Now, we have but R is asymmetric. Therefore, we have reached a contradiction and .

Are this all logically correct?