Given: The domain of the relation $R$ is $\{a, b, c, d\}.$ $R = \{(a, b), (b, a), (c, d), (d, c)\}.$
Why is this relation not transitive? According to my textbook,
By definition
R is transitive if and only if for every three elements, x, y, z ∈ A, if xRy and yRz, then it must also be the case that xRz.
and R is not transitive when
The situation that is not allowed in a transitive relation is for there to be an x, y, and z , such that xRy and yRz are true but xRz is not true. Notice that the definition of transitive is a universal statement. If any x, y, and z in the domain have the forbidden pattern of xRy and yRz but not xRz, then the relation is not transitive. If there is no triple x, y, and z that has the forbidden pattern, then the relation is transitive.
I thought it was transitive because it says "for every three elements", if $xRy$ and $yRz$ then $xRz$. Since there are no three elements where the situation $xRy$ and $yRz$ even occurs, I assumed it would be considered transitive.
Furthermore, the second excerpt states that "If there is no triple x, y, and z that has the forbidden pattern, then the relation is transitive." And as far as I can see, there is no triple, with the forbidden pattern.
I am confused.
Note that the definition does not say "for every three distinct elements". In the given example, with $x:=a$, $y:=b$, and $z:=a$, we see that $xRy$ and $yRz$, but not $xRz$ (i.e., $aRb$ and $bRa$, but not $aRa$).