Question: "Let $\mathscr R$ be an equivalence relation on a set $\mathcal A$ with exactly 4 equivalence classes, namely $\mathcal A_1$, $\mathcal A_2$, $\mathcal A_3$, and $\mathcal A_4$ such that $|\mathcal A_1| = |\mathcal A_2| = 10$ and $|\mathcal A_3| = |\mathcal A_4| = 5$. Determine $\mathscr R$."


This is a review question for my discrete math course, and I'm honestly not even sure where to start.

There's no knowledge given about the set $\mathcal A$ other than it has $30$ elements (obtained from the sizes of the equivalence classes), so I'm not sure how it is possible to determine anything about the relation $\mathscr R$.

I thought about relabeling the elements of $\mathcal A_1$ to $\mathcal A_4$ as $a_1, a_2,\ldots, a_{30} $, then determining the individual elements of $\mathscr R$, but that doesn't make sense as a solution to this question. Am I missing something?

I'm not looking for any solutions, just a push in the right direction.

  • $\begingroup$ Welcome to math.se. Here is a formatting guide. $\endgroup$
    – J.G.
    Dec 9, 2019 at 22:15
  • $\begingroup$ Are you sure the question is not "determine the number of relations satisfying these conditions?" $\endgroup$
    – J.-E. Pin
    Dec 9, 2019 at 23:30

1 Answer 1


R = $\cup${ A$_j$ × A$_j$ : j = 1,2,3,4 }.

Exercise. Show that there is a bijection between the equivalence relations of a set and the partitions of that set such that the equivalence classes of the relation are sets of the partition.


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