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Equivalence relations and the empty set

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Regarding the set of natural numbers N, answer the following questions.
Is â?? a binary relation? Explain.
Is â?? reflexive? Explain.
Is â?? symmetric? Explain.
Is â?? transitive? Explain.
Is â?? an equivalence relation? Explain.

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All the questions raised in this posting have been answered, and a justification has been given for each.

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A binary relation on N is a set of pairs (n, m) in N x N. The empty set has NO elements that are NOT of the form (n, m) in N x N, so this condition is satisfied (vacuously). Therefore, it IS a binary relation.

No, the relation is NOT ...

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Georgia Martin, PhD
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  • PhD, The Catholic University of America
  • PhD, The University of Maryland at College Park
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