Check whether the relation R ={(x, y)∈N×N | xy is the square of an integer} is an equivalence relation on N.
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Let us show that the relation is an equivalence relation on .
Sinse for any we have that , we conclude that , and hence the relation is reflexive.
If then for some It follows that and thus Therefore, the relation is symmetric.
If and , then and for some It follows that and hence We conclude that and the relation is transitive.
We conclude that the relation is an equivalence relation on .