Transitivity on a set

Question

I have a question about equivalence relationships. I know that a relationship is an equivalence relationship when the reflexive property, symmetric property and the transitive property exists on a set.

Ok...! ;-)

I have the following set

A = {0, 1, 2, 3, 4, 5, 6}

With the following relationship:

R = {(3, 3),(4, 0),(0, 0),(0, 4),(1, 1),(1, 6),(6, 1),(2, 2),(4, 4),(5, 5),(6, 6),(5, 3),(3, 5)}

With this data, we can draw the digraph:

Digraph

Then it's easy to check that the transitive property isn't exist in this case. is it true? and.., if is it true the question is... what kind of relationship is?

Answer 1

It is an equivalence relation, since it is reflexive, symmetric and transitive. If you think that it is not transitive, then please provide an example of elements $a,b,c\in\{0,1,2,3,4,5,6\}$ such that $(a,b),(b,c)\in R$, but $(a,c)\notin R$.

Answer 2

If you add for example (1, 0) and (0, 1) to R then it won't be transitive since (0, 1), (1, 6) ∈ R but (0, 6) ∉ R. That way it will not be an equivalence relation.