Is this a transitive relation

Question

I found this in a web site

If $A = \{ 1, 2, 3\}$, then the relation $R = \{(2, 3)\}$ is not transitive.

Why it is not transitive?

The definition is if whenever an element $a$ is related to an element $b$, and $b$ is in turn related to an element $c$, then $a$ is also related to $c$. In $R$ there is no matching pair for $(2,3)$, so transitivity can not be checked.

As the ordered pair in $R$ does not violate the condition, can't we say it is transitive?

Yes, you are right. It is transitive, and vacuously so. – Ken May 24 '15 at 15:51 — Ken, May 24 '15 at 15:51
Can you post the link to that web site? Thanks, – Gregory Grant May 24 '15 at 16:10 — Gregory Grant, May 24 '15 at 16:10

score 2 · Answer 1 · answered May 24 '15 at 16:00

It is indeed transitive. This is true vacuously; we need to check that if $(a,b)\in R$ and $(b,c)\in R$, then $(a,c)\in R$. But there are no two pairs $(a,b),(b,c)\in R$, so the required condition indeed holds for all such pairs (since there are none). I would prefer to say that transitivity can be checked, but there is nothing to check; not checking and checking the empty set are different things, for me at least.

Is this a transitive relation

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