Completeness from an example

Question

I have a set $X = \{1,2,3\}$ and a binary relation $B = \{(1,1),(1,2),(1,3),(2,3),(3,1)\}$.

I am trying to understand if this relation is complete.

The completeness definition I am using is if for each $x,y$ in $X$, either $xBy$ or $yBx$.

At first I thought $B$ was complete but both $(2,2)$ and $(3,3)$ are not in $B$. In the definition, if I set $x=2,y=2$ then I think we should have $(2,2)$ in $B$. On the other hand, it seems meaningless to me because isn't $2B2$ related with reflexivity? So for completeness do we actually need $(2,2) \in B$?

Answer 1

Yes, the relation $B$ over $X$ is not complete since $3 \in X$ but $(3,3) \notin B$.

The variables $x$ and $y$ in the completeness definition "$\forall x,y \in X, xBy \lor yBx$" are treated as placeholders and thus are not necessarily distinct. In fact, completeness implies reflexivity when you let $y = x$ in the definition.