Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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)

share|cite|improve this question
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
Do you mean "bounded" and "unbounded" by any chance? – Asaf Karagila Oct 10 '12 at 16:50
@AsafKaragila – Lc0rE 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

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 $M = 1$?

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.