From what I have read about a transitive relation is that if xRy and yRz are both true then xRz has to be true.
I'm doing some practice problems and I'm a little confused with identifying a transitive relation.
My first example is a "equivalence relation"
$S=\{1,2,3\}$ and $R = \{(1,1),(1,3),(2,2),(2,3),(3,1),(3,2),(3,3)\}$
My Book solutions say that this relations is
Reflexive and Symmetric
My Second example is "partial order"
$S=\{1,2,3\}$ and $R =\{(1,1),(2,3),(1,3)\}$
My Book solutions says is
Antisymmetric and Transitive
I got confused with why is the partial order(second example) Transitive. So what I did is that I applied $1R1$ and $1R3$ so $1R3$($xRy$ and $yRz$ so $xRz$).
I tried to applied this to my first example (equivalence relation). What I did is $1R1$ and $1R3$ so $1R3$ ($xRy$ and $yRz$ so $xRz$).
Can someone explain what I'm missing or doing wrong? What can I do to identify a transitive relation? As you can see on both practice examples both have the same set of relations $1R1$ and $1R3$ so $1R3$($xRy$ and $yRz$ so $xRz$) but one is transitive and the other is not.
Best Answer
To be transitive, $xRz$ needs to hold whenever you have $x$, $y$, and $z$ such that $xRy$ and $yRz$. You're only testing the case $x=y=1$ and $z=3$, but there might be other cases. For instance, in your first example, you can let $x=1$, $y=3$, and $z=2$, and you find that $1R3$ and $3R2$, but $1R2$ is not true. So the relation is not transitive.
(By the way, you should not call that first relation an "equivalence relation", since an equivalence relation is required to be transitive, in addition to being reflexive and symmetric.)