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:

Let R be a nonempty relation on A. If R is symmetric and transitive, then R is not irreflexive.

If this one was true then there wouldn't be equivalence relations...

Tonio
Since R is symmetric, assume

. Then, since R is transitive,

. Therefore, R is not symmetric.

Q.E.D.

If

, then R is anti-symmetric.

(1)

Let

. Therefore,

. By the anti-symmetric hypothesis, if

, then

.

(2)

Thus,

Hence, R is anti-symmetric.

If R is transitive and irreflexive, then R is asymmetric.

By contradiction: If R is transitive and irreflexive, then R is not asymmetric.

Let

. By transitivity,

. However, R is irreflexive. Therefore, we have reached a contradiction and R is asymmetric.

Are these all logically correct?