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## Answers

1) Every element in the set must appear in one of the equivalence classes, and

2) distinct equivalence classes have to be disjoint---meaning each term appears in exactly one equivalence class.

So, we can examine each of the sets in 1.--5. to make sure these hold.

1. Every element a, b, c, d, and e appears. There is only one set contained in that set, so each element only appears once. That's it. It has the required properties, so it could be the set of equivalence classes for an equivalence relation.

2. The element e doesn't appear in any of those subsets. So it can't be a partition.

The answers for the remaining three are 3. Yes, 4. No, 5. Yes. Just use the acid test described above to justify each of these answers.