Given any set A, a relation on A is a subset of AxA. Then isn't the empty set a relation also? Doesn't that make it an equivalence relation, vacuously, as well?(adsbygoogle = window.adsbygoogle || []).push({});

I'm asking because in a book there's a problem stating: show there are exactly 5 equivalence relations on a set with 3 elements. I get the obvious

{(1,1), (2,2), (3,3)}

{(1,1), (2,2), (3,3), (1,2), (2,1)}

{(1,1), (2,2), (3,3), (1,3), (3,1)}

{(1,1), (2,2), (3,3), (2,3), (3,2)}

{(1,1), (2,2), (3,3), (1,2), (2,1), (1,3), (3,1), (2,3), (3,2)} = AxA

But I think the empty set should also be included, because for example in {(1,1), (2,2), (3,3)}, symmetry and transitivity are both trivially satisfied, just as they would be in the empty set.

But I know equivalence relations correspond to partitions of the set. Then the partitions would be

{1} {2} {3}

{1,2} {3}

{1,3} {2}

{2,3} {1}

{1,2,3}

And the empty set doesn't partition A, so what should it be?

How is the empty set regarded with respect to (equivalence) relations?

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# Empty relation

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