# Example of an antisymmetric, transitive, but not reflexive relation

The question I'm tackling right now is this:

Give an example of a relation R on a set S that is not reflexive, transitive and not symmetric.

Let S = {1,2,3} and let R = {(1,1), (2,2), (1,2)}. Then R is irreflexive since (s,s) is not in R for every element s of S and R is not symmetric since (1,2) is in R but (2,1) is not in R.

I dunno how to answer the transitive part. Could you please assist me in this question?

Your help would be greatly appreciated.

• What about an empty relation? – dtldarek Dec 2 '14 at 0:17

To show that it is transitive, we have to show that if $(a,b)\in R$ and $(b,c)\in R$, then $(a,c)\in R$. In your example, $R = \{(1,1), (2,2), (1,2)\}$. So there are only two cases to check: $$(1,1)\in R\mbox{ and }(1,2)\in R,\mbox{ then }(1,2)\in R,$$ $$(1,2)\in R\mbox{ and }(2,2)\in R,\mbox{ then }(1,2)\in R.$$ Therefore, $R$ is transitive.
• To show $R$ is transitive, we just have to look at the elements in $R$. So $3$ in $S$ is irrelevant in this case. However, to show that $R$ is reflexive, you have to show that $(a,a)\in R$ for all $a\in S$. So $R$ is not reflexive because $(3,3)\not\in R$. – Paul Dec 2 '14 at 0:10