# Finding the transitive closure of a given relation on $X = \{1,5,7,9\}$

What is the transitive closure of the relation $$\{(1,5),(5,7),(7,9),(1,7),(1,9),(5,9)\}$$ on $$X=\{1,5,7,9\}$$?

I think this relation is already transitive.

• $$(1,5),(5,7) \implies (1,7)$$ already exist

• $$(1,5),(5,9) \implies (1,9)$$ already exist

• $$(1,7),(7,9) \implies (1,9)$$ already exist

Did I miss something or this relation is already transitive so my answer which is $$\{(1,5),(5,7),(7,9),(1,7),(1,9),(5,9)\}$$ correct?

• What is $A$ that is mentioned in the question? – D. Dmitriy Nov 4 '20 at 8:04
• On X this relation is already transitive indeed. – D. Dmitriy Nov 4 '20 at 8:05
• Oh typo it is X sorry and thank you for your answer! – Zeroo Nov 4 '20 at 10:17

You are indeed correct that the relation is already transitive. This is easy to see when you visualize the relation as a directed graph like below, where $$x$$ points to $$y$$ iff $$(x,y) \in R$$ (where $$R$$ is your relation on $$X$$).
In diagrams like these, transitivity amounts to, whenever you can travel from node $$x$$ to $$y$$ and then to node $$z$$, then there is a path letting you cut across from $$x$$ to $$z$$. Basically whenever two sides of a triangle exist, the third is also filled in (though the orientation of the third side depends on the other two).