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.
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...
Since R is symmetric, assume . Then, since R is transitive, . Therefore, R is not symmetric.
If , then R is anti-symmetric.
Let . Therefore, . By the anti-symmetric hypothesis, if , then .
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?