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Prove or disprove: every transitive relation on a set X with more 2 points is reflexive

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Let X be a set with more 2 points, and a\in X, b\in X, a\ne b. Consider the relation R=\{(a,a)\}\subset X\times X. Since (b,b)\notin RR is not reflexive. Taking into account that (a,a)\in R and (a,a)\in R imply (a,a)\in R for a unique element (a,a)∈R, we conclude that R is a transitive relation. Therefore, there exists a transitive relation which is not reflexive.

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Question ID: mtid-5-stid-8-sqid-3617-qpid-2316