# Transitive closure of a reflexive antisymmetric relation

Prove that the transitive closure $S$ of a reflexive antisymmetric relation $R$ is a partial order.

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Yes, this is obvious. We know the transitive closure must be transitive and reflexive as given, but the hard part is proving that it is antisymmetric. Can you help? –  Doug Jul 19 '13 at 2:09

## 1 Answer

The conjecture is false.

Let $A = \{1,2,3\}$ and let $R = \{(1,1),(2,2),(3,3),(1,2),(2,3),(3,1)\}$ which is reflexive and antisymmetric.

Its transitive closure, S = $\{(1,1),(2,2),(3,3),(1,2),(2,3),(3,1),(1,3),(2,1), (3,2)\}$, however is not antisymmetric since $(2,1)\in S$ and $(1,2)\in S$ but $1\neq 2$ and thus can't be a partial order.

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@amWhy $R$ is not transitive since $(1,2) \in R$ and $(2,3)\in R$ but $(1,3)\not\in R$. –  Alraxite Jul 19 '13 at 3:10
You're correct. Missed that pair. –  amWhy Jul 19 '13 at 3:12
Thanks Alraxite for your counter example that answers my question. –  Doug Jul 19 '13 at 12:56