What do the curly brackets in equivalence relations signify?

Question

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?

Answer 1

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.