R3 = {(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4), (3,1),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4)} Determine whether the relation R3 is reflexive, symmetric, anti-symmetric and transitive. Determine whether the relation R3 is an equivalence relation or partial order. Give reason for your answer
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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.