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Which of the following is the unbounded set ?

  1. $ X = \{ x\mid x = \frac{1}{n},n \in \mathbb{N} \} $

  2. $ Y = \{ x\mid x = \frac{1}{2^n},n \in \mathbb{N} \} $

  3. $ Z = \{ x\mid x = 2^n,n \in \mathbb{N} \} $

  4. $ W = \{ x\mid x \in \mathbb{N}, x \lt 4532 \} $

After reading the definition from Wikipedia , I can only convince myself that it can't be W but am not getting which is correct from the other three ?

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Think about What is the definition for a bounded set and an unbounded? –  user17762 Nov 10 '10 at 15:44
    
finite and infinite ? –  Quixotic Nov 10 '10 at 15:45
    
What is finite and infinite? –  user17762 Nov 10 '10 at 15:49
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Why did you have to go to wikipedia for a definition of bounded? The problem givers didn't provide one? Also, did you look at the Definition section in the wiki page you linked? –  Aryabhata Nov 10 '10 at 15:59
    
Next time you're writing sets in LaTeX, try this: $W = \left\lbrace x \in \mathbb{N} \; \middle\vert \; x < 4532\right\rbrace$. To see the source, just right click on the math and click "show source". And I do know that \middle doesn't work here, but it's in amsmath. –  kahen Nov 10 '10 at 16:03

2 Answers 2

up vote 2 down vote accepted

HINT: When do you say that a set is bounded.

  • A set $X \subset \mathbb{R}$ is said to be bounded if you can find a $M \in \mathbb{R}$ such that $|x| \leq M$ for all $x \in X$.

Now consider your cases:

  • $X = \{ x | x = \frac{1}{n} ; \ n \in \mathbb{N}\}$ *NOTE:* $\frac{1}{n} \leq 1$ for all $n \in \mathbb{N}$

  • Similarly the Second one.

  • Now , the third one is unbounded. (Why?). Assume $2^{n} \leq M$ for all $n \in \mathbb{N}$ and obtain a contradiction.

  • Fourth one follows from the definition.

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@Chandru1 I think you're missing an absolute value in your definition of a bounded set, or maybe you're just interested in saying it is bounded above. –  Adrián Barquero Nov 10 '10 at 19:23

A set in $\mathbb{R}$ is bounded if you can find two numbers, $a$ and $b$ such that all elements of the set are between $a$ and $b$. If you want to claim a set is unbounded, whatever $a$ and $b$ I give you, you should be able to find at least one element outside the interval $(a,b)$. Which of your sets satisfies that?

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Thanks a lot for explaining the concept :) –  Quixotic Nov 10 '10 at 18:17

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