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 is a subset of S?
I understand this question has been asked before and answered by several users. However, I am not convinced the answers of 15 given the "merging" of equivalence classes is correct in general. Consider the equivalence relation congruence modulo 4 on the set of integers. This relation has four distinct equivalence classes but the set of integers can have an infinite amount of equivalence relations, S, for which R can be a subset of any if we take R to be the relation that consists of only ordered pairs of the form (x, x) for all integers.
I would truly appreciate clarification on this point. Perhaps, I am thinking about this incorrectly but after considering any equivalence relation on the set of integers can contain all elements of R as described above in addition to any number of other elements that maintain the properties of being symmetric and transitive, It seems unlikely to me that the correct answer is 15 and not infinite for an infinite set. It would make sense if the set A is finite and thus, given 4 distinct equivalence classes of R, must have four distinct elements since R is an equivalence relation. But, otherwise, these answers seem wrong.