Number of reflexive, anti-symmetric, non-transitive relations on a set

Question

How many relations are there that are reflexive, antisymmetric and non-transitive are on 3-element set?

Answer 1

Let the set be $\{a,b,c\}$, and let $R$ be the relation.

Reflexivity gives $xRx$ for any $x$. In order for $R$ to be non-transitive, there must be a chain $xRy,yRz$, where $x,y$ and $z$ are distinct. Now there is not much freedom for $R$. $yRx$ and $zRy$ are ruled out because of antisymmetry. $xRz$ would exclude $zRx$ and make $R$ transitive. Therefore, we can either have $zRx$ (1) or not $zRx$ (2), and that completely determines $R$.

Case (1): There are 2 possibilities, $aRb,bRc,cRa$ or $aRc,cRb,bRa$.

Case (2): There $3!=6$ ways to choose $x, y$ and $z$ from $\{a,b,c\}$.

So we have $2+6=8$ solutions.