# Completeness from an example

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 $$(2,2) \in B$$?

• You did indeed find a violation of completeness. Actually, completeness implies reflexivity. – Michael Greinecker Oct 2 '18 at 13:59

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.