Discrete Math: Does all vertices need self-loops for a relation to be transitive?

[Cold01]

For example: Is the relation (x, y) ∈ { (1, 1), (1, 2), (2, 3), (1, 3) } transitive or would it need (2, 2) and (3, 3) to be considered transitive?

 

[Dr.Peterson]

For example: Is the relation (x, y) ∈ { (1, 1), (1, 2), (2, 3), (1, 3) } transitive or would it need (2, 2) and (3, 3) to be considered transitive?

Can you explain why it would need them? In what way does it not fit the definition of transitive as it is?

 

[Cold01]

Can you explain why it would need them? In what way does it not fit the definition of transitive as it is?

Never mind, you're right. Self-loops aren't directly related to if a relation is transitive or not.

I was a bit confused because a lot of the examples I've viewed always represented transitive relations with a graph that either had no self-loops or self-loops for every vertex. So I got curious if they were also connected to a relations transitivity.

Thank you, I'll mark the question as solved.

 

[lex]

For example: Is the relation (x, y) ∈ { (1, 1), (1, 2), (2, 3), (1, 3) } transitive or would it need (2, 2) and (3, 3) to be considered transitive?

The relation R ={(1,1), (1,2), (2,3), (1,3)} is transitive.
(R a relation on an appropriate set).

A relation R on a set A, is transitive [MATH]\hspace2ex \leftrightarrow \hspace2ex a, b, c \in A \rightarrow [\hspace1ex \left(\,(a,b)\in R \text{ and }(b, c) \in R \, \right) \rightarrow (a, c) \in R \hspace1ex][/MATH]which is clearly true for the elements of your relation.
It seems you are confusing it with reflexivity.

 

[Dr.Peterson]

Never mind, you're right. Self-loops aren't directly related to if a relation is transitive or not.

I was a bit confused because a lot of the examples I've viewed always represented transitive relations with a graph that either had no self-loops or self-loops for every vertex. So I got curious if they were also connected to a relations transitivity.

Thank you, I'll mark the question as solved.

That's one problem with learning from examples (though of course they are necessary): we can't give examples of every situation. Probably most of the examples you've seen are intended to show equivalence relations, so you haven't seen enough transitive relations that are not also reflexive.

I was wondering if you were thinking backward in some way, or perhaps misread the relation and thought that (2,2) was implied.