Let S be a finite non-empty set. How many relations on S are simultaneously an equivalence relation and a partial order? Justify your answer.
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be a relation that is simultaneously an equivalence relation and a partial order, that is is reflexive, transitive, symmetric and antisymmetric. Since is reflexive, for any Let . Since is symmetric, we conclude that . Taking into account that is antisymmetric and , we conclude that Therefore, is the unique relation.