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I tried to prove that if $R$ is transitive then its inverse is transitive as well. $\begin{align} & a{{R}^{-1}}b\,\,,\,\,b{{R}^{-1}}c \\ & \Rightarrow bRa\,\,,\,\,cRb \\ & \Rightarrow cRa \\ & \Rightarrow a{{R}^{-1}}c \\ \end{align}$

is this a correct proof (or am I completely wrong and it's not even true?)

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A note: What you call "complementary" is conventionally called the inverse relation of $R$. (It's interesting that you got the notation right.) Wikipedia says that alternative terms are converse or transpose relation. I am not sure if "complementary" is used by anyone.

True statement and correct proof.

This statement from wikipedia article on inverse relation is relevant:

If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, its inverse is too.

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    Hehe, such a discussio$n$ about my i$n$correct tra$n$slation in a hurry from my language :)2011-07-31
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Yes, that proof looks fine to me.

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    And Thanks to you 2 :)2011-07-30