Hello zhupolongjoeAs you have written the question, some of it doesn't make sense. I'll explain why, but it will take a little time, so stay with me.

, the power set of , is theset of all the subsetsof . So:

Look carefully at the words above in italics: is aset of ... subsets ...; so the elements of are themselves sets - and I have listed a few of them above.

Now is a relation on (not a relation on ) - in other words is a relation betweentwo subsets of, which is: subset is related to subset if and only if and have the same number of elements. So, for example:

- because they both have 2 elements
- because they both have 3 elements

... and so on.

Next, you need to understand the notation. If a relation is defined on a set , then (called thequotient setof by ) is the set of all the equivalence classes into which is 'divided' (orpartitioned) by (hence the use of the word 'quotient').

For example, if and (in other words, is divisible by ), then partitions into two equivalence classes: {even integers} and {odd integers}, and these two sets are therefore the elements of .

So, in the question you're asking, whereas it makes sense to talk about , it makes no sense to talk about or , because is not defined on individual elements like , but onsets.

OK. So let's look at the question that does make sense: How many elements are there in ? This question means: when is partitioned into equivalence classes by , how many of these equivalence classes are there? Well, for example, all the subsets of that contain one element - that's - are equivalent. So they form one equivalence class. (And that, I suspect, is what the question means when it says 'list all the elements in ', although that is incorrect use of the notation.)

All the subsets of containing 2 elements form another equivalence class; all the subsets containing 0 elements form another, and so on. Since you can form subsets containing 0, 1, 2, 3, 4, 5 or 6 elements, there are therefore 7 equivalence classes, and therefore 7 elements in .

The proof that is an equivalence relation is trivial. Any relation that contains the words ' ... the same as ' usually is. If has the same number of elements as , then it's obvious that (reflexive) and (symmetric) and (transitive).

Grandad