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I am trying to study binary relations (for myself, it's not an assignment!) I have the set $\{1,2,3,4\}$, and one of the relations in the exercise is $\{(1,3),(1,4),(2,3),(2,4),(3,1),(3,4)\}$.

A relation is called transitive if whenever $(a,b)$ is a member of $R$ and $(b,c)$ is a member of $R$ then $(a,c)$ is also a member of $R$.

Book says it's not a transitive relation. Why?

In given relation we have $(2,3)$, $(3,4)$ and $(2,4)$. ($x = 2$, $y = 3$, $z = 4$)

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    $\begingroup$ Observe that you have $(2,3)$ and $(3,1)$ but you don't have $(2,1)$. The relation is transitive only if it contains $(x,z)$ whenever it contains both $(x,y)$ and $(y,z)$. $\endgroup$ – MJD Oct 20 '14 at 21:51
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Observe that you also have (1,3) and (3,1). However you don't have (1,1).

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  • $\begingroup$ Of course! Okay, but can you tell me why is this relation transitive ? {(1,1),(2,2),(3,3),(4,4)} $\endgroup$ – Jenkins90 Oct 20 '14 at 21:51
  • $\begingroup$ This is the equality relation of the set $\{1,2,3,4\}$. Equality is of course transitive! $\endgroup$ – Berci Oct 20 '14 at 21:52
  • $\begingroup$ @Jenkins90 That relation is transitive because whenever it contains $(x,y)$ and $(y,z)$, it also contains $(x,z)$. $\endgroup$ – MJD Oct 20 '14 at 21:52
  • $\begingroup$ They didn't say anything about equality relations in this chapter.. :<. Thanks a lot for your help! $\endgroup$ – Jenkins90 Oct 20 '14 at 21:53

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