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Bleys
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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?
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?
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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