# Transitive closure clarification

I know what it means by the term. However, if I have a relation R = {(1,2),(4,5),(2,3)} on a set A = {1,2,3,4,5}.

I know that straight away by drawing that (1,3) is a transitive edge. But with (4,5) to make the whole relation transitive can we draw an edge (3,5). i.e.

(1,2) and (2,3) -> (1,3) (3,4) and (4,5) -> (3,5) (2,3) and (3,4) -> (2,4) (2,4) and (4,5) -> (2,5) (1,3) and (3,4) -> (1,4) (1,4) and (4,5) -> (1,5)

This would mean R' = {(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)}

or is the best way without drawing an extra edge:

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

Thanks.

The correct answer is $$\{(1,2), (1,3), (2,3), (4,5)\}.$$
I can't say I understand your motivation for adding more edges than that. You seem to say the presence of $$(3,4)$$ and $$(4,5)$$ implies that $$(3,5)$$ needs to be there, but that's wrong since $$(3,4)$$ isn't present. It wasn't there in the first place and you didn't need to add it in the previous step where you (correctly) added $$(1,3)$$. So you don't need to add $$(3,5).$$
• @FraserGilbert No, I don't think you're understanding. Every graph we consider has $5$ vertices. We are just adding edges. The fact that vertices $4$ and $5$ are disconnected from the others doesn't matter. I still don't understand your initial inclination to connect them. My wild guess is that the word "transitive" sounds like it might mean "we can get from one side to the other", and you're assuming it means or implies that. If that's it, erase that association in your brain cause that's nothing remotely close to what "transitive" means. Reread the definition and think about it. Sep 1, 2020 at 23:20