# Transitive Relations on a set

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 $(2,4)$. ($x = 2$, $y = 3$, $z = 4$)

• 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 $(y,z)$. – MJD Oct 20 '14 at 21:51

• This is the equality relation of the set $\{1,2,3,4\}$. Equality is of course transitive! – Berci Oct 20 '14 at 21:52
• @Jenkins90 That relation is transitive because whenever it contains $(x,y)$ and $(y,z)$, it also contains $(x,z)$. – MJD Oct 20 '14 at 21:52