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?
 A: Yes, and yes, respectively. For example, the relation $\{(a,b),(b,a)\}$ is not transitive; its transitive closure is $\{(a,a),(a,b),(b,a),(b,b)\}$. So the useful information you get from knowing that you have a transitive relation containing $(a,b)$ and $(b,a)$ is that you must also have $(a,a)$ and $(b,b)$.
A: 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.
A: It may or may not be transitive. It depends on the set and the relation.


*

*Let S={} then R={} and it is trivially transitive.

*Let S={a} then there are two relations R1={}, R2={(a,a)} and both are transitive trivially.

*Let S={a,b} then there are 16 relations


*

*R1={}                     -->  Transitive

*R2={(a,a)}                -->  Transitive 

*R3={(b,b)}                -->  Transitive

*......

*R6={(a,a),(a,b)}          -->  Transitive

*R7={(a,b),(b,a)}          -->  Not Transitive

*and so on.


A: Recall the definition of transitivity : for all x,y,z in a set A with the relation R, if xRy and yRz then xRz.
Now, if we have 2 elements, just think to yourself that our condition can't be met (unless you take 2 elements to be equal), and whenever the conditions of a property can't be met then we consider the property to hold.
Mathematically, it's not entirely correct (I get that) but it's a good way to understand it simply I find.
