O.K. I mean by the word group to set. Sorry. I still didn't understand why it a class and isn't set.
Why the rational set is a class and is not set?
Words are important! And words in mathematics are given very careful definitions that are different from their ordinary usage, as JeffM well said. We commonly take some vague word and assign it a much more specific meaning. Often the choice is ultimately random.
If you are talking about a textbook in English, you need to be using the words used there, as they define them. If you are talking about a textbook in Hebrew, you have to use their words, and be very careful about translating them to English when asking questions about them. This can be very tricky; what you think is a synonym may not be.
To a mathematician, a
group is a set associated with an operation with specific properties; it is not the same thing as a set.
The word "
set" is very general, and refers to any collection of anything (though it is given a precise definition in some branches of set theory).
The word "class" doesn't, in general, have such a precise meaning (except
sometimes); but the phrase "
equivalence class" does. Why was the word "class" chosen there rather than "equivalence set"? Who knows? That's just the way it is; there doesn't have to be a reason. (And, of course, in other languages, other words are used just as arbitrarily!)
But please observe that the
rational numbers are a
set of numbers; each
rational number, defined as you describe, is itself an
equivalence class of ordered pairs, which is not quite the same idea as a mere set. Are you just being lazy when you say "
rational set", and not taking the trouble to say what you mean, or are you confused about this difference?
