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I'm having some trouble with the definition of "Bounded Set". I have a pretty good idea of what "Limited" means: a Set with a Upper and a Lower bounds. Now i have a quiz in which I must choose the right answer and I have absolutely no idea what to chose:

With A ⊆ R and M ∈ R+, A is Limited if:  (a) ∀M ∈ R+ : ∃a ∈ A : |a| > M (b) ∃a ∈ A : |a| > M, ∀M ∈ R+ : (c) ∃M ∈ R+ : |a| ≥ M, ∀a ∈ A (d) ∃M ∈ R+ : ∃a ∈ A : |a| > M (e) ∀M ∈ R+ : |a| ≥ M, ∀a ∈ A 

In the same way:

With A ⊆ R and M ∈ R+, A is Unlimited if:  (a) ∀M ∈ R+ : ∃a ∈ A : |a| > M (b) ∃M ∈ R : ∃a ∈ A :|a| > M (c) ∀a ∈ A : ∃M ∈ R+ :|a| ≥ M (d) ∃M ∈ R+ : |a| ≥ M, ∀a ∈ A (e) ∀M ∈ R+ : |a| ≥ M, ∀a ∈ A 

Can you chose the right answer? ( I have the solutions of course but i want a clear explanation of what an limited and unlimited set is). Thanks

Edit: the right answers: (c) and (a)

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    You should interpret the sentences (read $\forall$ as '*for all*' and $\exists$ as '*there exists.. such that*').. Anyway. what is the solution?2012-10-10
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    Do you mean "bounded" and "unbounded" by any chance?2012-10-10
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    @AsafKaragila http://en.wikipedia.org/wiki/Upper_and_lower_bounds2012-10-10
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    And nowhere on that page there is any use of the term "Limited set".2012-10-10
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    The directions of the relations are everywhere the other way around.. Should be $|a| or $|a|\le M$ in the winner formulas (supposed that 'limited'='bounded'). Can you check them?2012-10-10
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    @AsafKaragila I'm sorry, I mean "Bounded Set" http://en.wikipedia.org/wiki/Bounded_set2012-10-10

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