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Consider the relation on the set
- Since for each the relation is reflexive. Taking into account that implies for any pair we conclude that the relation is symmetric. Since and this relation is not anti-symmetric. Taking into account that and implies for any pairs we conclude that the relation is transitive.
- Since the relation is reflexive, symmetric and transitive, it is an equivalence relation. Taking into account that is not anti-symmetric, we conclude that is not a partial order.