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I don't really have anything to add to the comments Ron made above, but I don't find the question at all ambiguous. I do, however, think these questions can seem abstract if you haven't seen a similar question before. As for this:

there are two very similar questions to this one in the Official Quant Review book (the green book) - Q158 in the PS section, and Q70 in the DS section.

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so i think this is the problem here -- stated in two different ways:
1/ you are assuming without cause that a and b stand for DIFFERENT integers;
or, alternatively,
2/ you are assuming without cause that the set contains at least two different numbers (or four, or whatever you're saying).

neither of these is true; in mathematics, when we say “if a and b belong to a set…”, weirdly enough, the default is to allow the situation in which a and b both stand for the same element of the set.
for instance, if s contains absolutely no numbers except 0, it actually satisfies both of the postulates required here: (i) we know that 0 is in S, and -0 = 0 is also in S; (ii) the only possible a and b in this statement are 0 and 0, and indeed ab = 0 is also in the set.

i can totally understand your confusion here, because this is so weird compared to the way things are normally done in the real world.
for instance, if someone says “two people x and y are standing in a park…”, then this person is obviously assuming that x and y stand for different individuals. however, that's just not the way that mathematics is done, by convention -- the default assumption is that different letters may, in fact, stand for the same number.
as a side note, this is why you see so many problems that explicitly mention “different integers” (because this is NOT assumed unless you actually say it) -- but why you will never ever see a problem that says “two integers which could be the same” (because this is already assumed).

hope this helps.

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Ron is a Director of Curriculum Development at Manhattan GMAT. He has been teaching various standardized tests for almost 20 years.

He wears white after Labor Day, gets 55% of his calories from protein, and takes standardized tests for fun.

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by the way, here's another GMAT PREP problem:

this problem can absolutely be re-written as follows:

In the sequence a(sub 1), a(sub 2), a(sub 3), ... [these are subscripts], ..., aN, is each element of the sequence equal to 0?
(1) For any integers u and v, where 1 < u < N and 1 < v < N, the product of a(sub u) and a(sub v) is 0.
(2) For any integers u and v, where 1 < u < N and 1 < v < N, the sum of a(sub u) and a(sub v) is 0.

do you see the connection?

_________________
Ron is a Director of Curriculum Development at Manhattan GMAT. He has been teaching various standardized tests for almost 20 years.

He wears white after Labor Day, gets 55% of his calories from protein, and takes standardized tests for fun.

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amit, the question is not ambiguous -- the problem lies in your understanding of the notation and mathematical language, not in the question statement. i know this may be somewhat of a blow to your ego, but you are not going to learn anything if your modus operandi is to continue denying the reality of how mathematical notation is used. are you on this forum to learn, or to argue?

i'm going to try to explain this in one more way, and then i will let this thread rest.
if the notation is confusing you, here's what these statements mean without the notation:

--> this means, "for any number that is in S, the opposite of that number is also in S."
this statement does NOT reference a particular value of "a"; it is a general truth about all values in the set.

--> this means, “for any two numbers that are in S (even if that's the same number twice), the product of those numbers is also in S.”
again, this statement does NOT reference particular values of "a" and "b"; it's another general truth about all values in the set.

this is how this sort of notation works.
full disclosure: i am a mathematician by formal training; i have studied with books and professors from all around the world, and this is how this kind of notation is used absolutely everywhere. so, you should take a moment to
(a) learn the correct meaning of these statements, and
(b) write your own examples of situations and statements that work in the same way -- this is the best way to learn notation/language with which you are unfamiliar.

good luck.

_________________
Ron is a Director of Curriculum Development at Manhattan GMAT. He has been teaching various standardized tests for almost 20 years.

He wears white after Labor Day, gets 55% of his calories from protein, and takes standardized tests for fun.

Pueden hacerle preguntas a Ron o en inglés o en español.

If you send Ron a private message, please allow 1-2 weeks for a response.

Learn more about ron