Equivalence relation $\mathscr R$ from the size of equivalence classes

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 $$\mathscr R$$."

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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.

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Dec 9, 2019 at 22:15
• Are you sure the question is not "determine the number of relations satisfying these conditions?" Dec 9, 2019 at 23:30

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