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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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1  
You should interpret the sentences (read $\forall$ as 'for all' and $\exists$ as 'there exists.. such that').. Anyway. what is the solution? –  Berci Oct 10 '12 at 16:43
2  
Do you mean "bounded" and "unbounded" by any chance? –  Asaf Karagila Oct 10 '12 at 16:50
    
@AsafKaragila en.wikipedia.org/wiki/Upper_and_lower_bounds –  l_core Oct 10 '12 at 16:58
    
And nowhere on that page there is any use of the term "Limited set". –  Asaf Karagila Oct 10 '12 at 17:06
    
The directions of the relations are everywhere the other way around.. Should be $|a|<M$ or $|a|\le M$ in the winner formulas (supposed that 'limited'='bounded'). Can you check them? –  Berci Oct 10 '12 at 17:06

1 Answer 1

For the first one, you could ask your self which of the statements are true for $A= \mathbb{R}$ itself (assuming here that $R$ is the real numbers.

For example the first statement if true for the real numbers since indeed for any real number $M$ ($\forall M$ ) you can find another number $a$ ($\exists a$) such that the absolute value of $a$ is greater than $M$ ($\lvert a \lvert > M$). This shows that the first statement does not say that $A$ is bounded/limited since the example with $A=\mathbb{R}$ is unbounded.

Try to analyse each statement like that and remember that the empty set is bounded.

And oops, I think I just gave away the answer to the second problem.

Edit: You indicate in your question that you believe that (c) is the correct answer for the first problem. Try to consider the set $A = [2,\infty)$. Then $A$ is unbounded, but does not the statement hold with $M = 1$?

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