Why is this Relation R (graph) not transitive?

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Let the arrow graph of R be the following:

enter image description here

If we get the ordered pairs we have that

R = { (a,a), (a,b), (a,c), (b,b), (b,a), (b,c), (c,a), (c,b), (d,d) }

If we analyze this:

*Reflexive - NOT TRUE. Because there aren't x = x for every ordered pair in the set

*Symmetric - TRUE. Because for every (x,y) there is a (y,x)

Question: Transitive?

My professor said that this was not transitive. I am wondering why.

With the pairs we can see that for every (x,y) -> (y,z) there is indeed a (x,z) pair.

Could you please tell me why this is NOT TRANSITIVE?

Thank you!

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3 Answers

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You say:

With the pairs we can see that for every $(x,y)\to (y,z)$ there is indeed a $(x,z)$ pair.

Are you sure? What about $c,a,b$?

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$R$ does not contain $(c,c)$ and this while $cRa\wedge aRc$. This contradicts that $R$ is transitive.

In general a relation $S$ is transitive if $uSv\wedge vSw$ implies that $uSw$.

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Hint: Look at $(c,a)$ and $(a,c)$.

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