Studying relation properties. My definition of a transitive relation is as follows:
A relation is transitive if and only if $\forall a,b,c \in A [aRb \land bRc \implies aRc]$
My question is: if $aRb \land bRc$ never occurs in the first place, is the relation considered transitive?
I ask this because when I read $\forall a,b,c \in A [aRb \land bRc \implies aRc]$ I read it as follows: "when $a$ relates with $b$ and $b$ relates with $c$, $a$ relates to $c$". But since this "when" never happens, there is no condition to evaluate to decide if it is transitive or not.
The same issue occurs with antisymmetry, where $\forall a,b \in A[aRb \land bRa \implies a = b]$ - what if $aRb \land bRa$ never occurs in the first place?
I just remembered that $F_0\implies whatever$ is always $V_0$.... I guess this answers my question... kinda.