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?