relations and trichotomy

Question

Is it true if there is a relation $R$ on A and $(x,x) \in R$ for some $x \in A$ then the relation does not have trichotomy?

For instance if $A=\{a,b,c,d\}$

$R=\{(d,c),(c,a),(b,d),(d,a),(a,a),(b,c),(b,a)\}$ does $R$ not have trichotomy since $(a,a) \in R$?

Or does it have trichotomy?

I was just asked the following question.

(T/F) If $(x,x)\in R$ for some $x \in A$ the $R$ does not have trichotomy.

The answer was true and I am confused?

Answer 1

$R$ is trichotomous iff for any $x$ and $y$: exactly one of $xRy$, $yRx$, or $x=y$ holds.

So, if you pick $x=y$, that means that for any $x$: exactly one of $xRx$ or $x=x$ holds.

But of course we always have $x=x$, and so we can never have $xRx$ if $R$ is trichotomous.