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 $(2,2) \in B$?

  • 2
    $\begingroup$ You did indeed find a violation of completeness. Actually, completeness implies reflexivity. $\endgroup$ Oct 2, 2018 at 13:59

1 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.