Can a relation with less than 3 elements be considered transitive? The generalize rule for a transitive relation is 
a -> b
b -> c
therefor
a -> c

If an element has less than 3 elements, can it still be transitive? If so, does that provide any useful information?

The general rule for a relation $\;\sim\;$ to be a transitive relation on a set $S$, we must have that for all $a, b, c \in S$, with $a, b, c\;$ not necessarily distinct.


*

*IF   $\;a \sim b\;$ AND IF  $\;b \sim c,$

*THEN we MUST have that $a \sim c$


If there happens to be less than three elements, then provided reflexivity holds for all $a \in S$, and symmetry holds for all $a, b \in S$, then transitivity follows. 
Say we have the relation $R$ denoted by $\sim$ on the set $\{a, b\}$.$\,\,\,$
Then provided $a \sim a$, $b \sim b$, $a \sim b$ AND $b \sim a$, so that $R =\{(a, a),(b, b), (a, b), (b, a)\}$, then $R$ is reflexive, and symmetric, and must therefore be transitive, given there are only two elements. 
If $S = \{a\}$, then any relation that is reflexive, i.e. any relation for which $R = \{(a, a)\}$ happens also to be (trivially) symmetric and transitive.

The only time transitivity fails is when there exists a, b, c such that 
$a \sim b$ and $b \sim c$, BUT NOT $a \sim c$.
Sometimes it's easier to understand that a relation is transitive, UNLESS there exists a counterexample as described immediately above.
