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Consider the set of words {one, two, three, four, five, six, seven, eight, nine, ten}. I want to write the associated partitions of this set under the relation R where two words are equivalent if they have the same number of letters. It's obvious one, two, six, and ten are all equivalent but I'm not exactly how to write the partitions with the curly brackets... Would it be {{one, two, six, ten}, {four, five, nine}, {three, seven, eight}}? Or would I write these partitions separately like {{one},{two},{six},{ten}}, {{four},{five},{nine}}, and {{three},{seven},{eight}}? What do the curly brackets actually say and what's the difference between the two ways I just wrote them?

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  • $\begingroup$ The curly braces define a set. The first way you wrote it is closest to correct. You have broken your original set into three partitions. Each one of those is a set and the union of all three of the sets equals the original set. Therefore, your first partition would be $\{\text{one}, \text{two}, \text{six}\, \text{ten}\}$. $\endgroup$ – John Douma Nov 27 '16 at 19:44
  • $\begingroup$ @JohnDouma So what is the second way I wrote it saying? From what you said I would guess it means {one}, {two}, {six}, and {ten} are each their own equivalence classes. But what does it mean when I group them into curly brackets like this: {{one},{two},{six},{ten}}? $\endgroup$ – Bryyo Nov 27 '16 at 19:48
  • $\begingroup$ The last thing you wrote is a set of four one element sets. $\endgroup$ – John Douma Nov 27 '16 at 19:50
  • $\begingroup$ A partition of any set $S$ is just a collection of subsets of $S$ such that the union of the collection is equal to $S$ and none of the individual subsets in the collection intersect. There is no special notation needed for this. $\endgroup$ – John Douma Nov 27 '16 at 19:52
  • $\begingroup$ @JohnDouma Ah, okay. I think that kind of makes sense. So when I wrote "{{one},{two},{six},{ten}}, {{four},{five},{nine}}, and {{three},{seven},{eight}}", would this imply that these are three different ways to partition the set? But basically all three partitions are equivalent because the elements inside would be the equivalent classes under the same relation R? Obviously this is wrong but is this what it would be saying? $\endgroup$ – Bryyo Nov 27 '16 at 19:56
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Curly brackets are used to specify sets. For example, the set of days in a week is {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}.

The partitions of a set under an equivalence relation are called the equivalence classes of the relation. These equivalence classes form a partition of the set. And the partitions too are sets. So if you have to write the set of the partitions, you would write them as your first notation. Your second notation doesn't make any sense in regard to equivalence classes.

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