Question

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

Answer 1

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 $(b,b)\notin R$, $R$ 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.