2
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ Yes, you are right. It is transitive, and vacuously so. $\endgroup$ – Ken May 24 '15 at 15:51
  • $\begingroup$ Can you post the link to that web site? Thanks, $\endgroup$ – Gregory Grant May 24 '15 at 16:10
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.