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Say I want to find the number of equivalence relations on the integers containing some other relation: specifically, if I have $aRb$ whenever $ a = b + 5$, how do I find the number of equivalence relations containing it? Obviously mod 5 is one, but how do I find others? Aren't their infinitely many?

By contains I mean: The relation $S$ contains the relation $R$ if $aSb$ whenever $aRb$

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    Don't you need R to be reflexive, symmetric and transitive? – 2017-01-09
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    Not R in this case: but the relation "containing it", yes – 2017-01-09
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    $\mod 5$ is the smallest equivalence containing $R$. Now, larger equivalences simply are constructed by any equivalence on $\Bbb{Z}/5 \Bbb{Z}$: this set has 5 elements, and so the number of equivalences on it is the fifth Bell number: 52. – 2017-01-09
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    Crostul I'm having trouble seeing how that would work, could you provide an example? – 2017-01-09

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