Transitive and Antisymmetric Relations

Transitive relation

A relation $R$ on a set $A$ is a transitive relation if for any two ordered pairs $(a,b)\in R$ and $(b,c)\in R$, then the ordered pair $(a,c) \in R$. The relation $R$ is not transitive if there are ordered pairs $(a,b)\in R$ and $(b,c)\in R$ so that $(a,c)\not\in R$.

Let’s take the set $M= \{\alpha, \beta, \gamma\}$ to see some examples. The following relations on $M$ are examples of transitive relations:

Now, let’s look at the relation $R_8$ shown in the image below to see why it’s transitive. Because $(\alpha , \beta)\in R_8$ and $(\beta , \gamma)\in R_8$, $(\alpha, \gamma)$ should be in $R_8$, and we observe that $(\alpha, \gamma)\in R_8$. Similarly, because $(\alpha , \beta)\in R_8$ ...