Suppose $R$ is an equivalence relation on a set $A$, with four equivalence classes. How many different equivalence relations $S$ on $A$ are there for which $R⊆S$? Thanks in advance
2026-04-09 02:21:20.1775701280
How many equivalence relations $S$ on $A$ are there for which $R⊆S$ ($R$ is an equivalence relation on a set $A$, with $4$ equivalence classes)
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Since R is a subset of S the equivalence classes can be the same as R or we can merge a few together. We can have S=R (1 case) or two equivalence classes merged(6 cases), or two merged and another two merged (6 cases) or three merged(4 cases) or all of them merged together (1 case).