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?

Solution 1:

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.

Solution 2:

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 $(b,b)$.

Solution 3:

It may or may not be transitive. It depends on the set and the relation.

1. Let S={} then R={} and it is trivially transitive.
2. Let S={a} then there are two relations R1={}, R2={(a,a)} and both are transitive trivially.
3. 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.