Transitive, Symmetric and Antisymmetric for relations.

Question

Determining whether a relation is reflexive is straightforward, as it is for symmetric and transitive.

But considering how strict the criteria, does it mean that any relation that simply fails the (P) part of the P then Q equation is vacuously transitive/symmetric/antisymmetric?

Example: For transitivity: for all $a, b, c \in X$, if $a R b$ and $b R c$, then $a R c$.

Let $A =\{a,b,c,d\}$, let $R$ on $A$ be $\{(a,a)\}$. There's not even a second element so that I could infer a relation, but would I call this relation vacuously transitive, or something else entirely?

Answer 1

Sure, a relation $R$ is transitive if for all $x,y,z$, if $(x,y) \in R$ and $(y,z) \in R$, then $(x,z) \in R$. If $R$ is empty, say, then it is trivially transitive. I don’t think it has a name since it is not very interesting…

Answer 2

Note that the definition of transitivity does not require that $a,b,c$ be distinct, so the "if" part (antecedent) is satisfied for $a=b=c$ in your example of $R$.

A vacuous truth is one where the antecedent is always false. In that sense, I suppose you could call an empty relation "vacuously transitive."