Set Proof

Proof that number of partitions on a set = number equivalence relations?

“By considering partitions, calculate the number of equivalence relations on a set with 4 elements.”

The equivalence classes of an equivalence relation form a (unique) partition of the set. Thus, each partition gives rise to an equivalence relation:

e.g. {1,4} {2} {3} means the equivalence relation is

{(1,1), (1,4), (4,1), (4,4), (2,2), (3,3)}.

For your second equation, I’m not sure, but you might have to subtract off the “empty” relation where nothing is related at all with in the set.

For reflexive, we need (i,i) for i between 1 and n. Thus, the decision for those has been made. The rest we have a choice for. Answer: 2^(n^2 – n).

For symmetric, we can impose an order on the elements. Then we only have a choice for (a,b) where a b, then the decision is made, that being the choice made for (b,a)). Answer: 2^(1+2+3+…+n) = 2^(n(n+1)/2).

Hope this helps.

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