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.
My answer:
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.
Best Answer
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.