What would a counterexample look like to prove the union of two transitive relations on the same set need not be transitive?
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On the set $\displaystyle \left\{ {a,e,i,o,u} \right\}$ both $\displaystyle R = \left\{ {\left( {a,e} \right)} \right\}\,\& \,S\left\{ {\left( {e,i} \right)} \right\}$ are transitive.
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