# Thread: What does this notation imply?

1. ## What does this notation imply?

See attached image.

I am not sure of how to use the bracketed numbers [0, 1-1/n] for each case.

Any help appreciated,
thanks

2. ## Re: What does this notation imply?

first is simply taking the infinite union of different intervals

second is infinite intersections of similar interval ranges.

3. ## Re: What does this notation imply?

I understand that, although not what is meant with the added [0, 1-1n]

4. ## Re: What does this notation imply?

it's a closed interval.

5. ## Re: What does this notation imply?

Still not entirely sure what you mean,
would you care to be more specific?

6. ## Re: What does this notation imply?

consider case when n=1, we have a degenerate closed interval, limiting n to infinite creates a set from [0,1]

7. ## Re: What does this notation imply?

Hi again,
I don't mean to come off as rude or ungrateful, as you are offering me help out of your spare time for nothing.
However, I get the feeling you are being vague on purpose with your replies.

If that is not the case, then I apologise, I am somewhat frustrated. If it is the case, can you please stop trying to confuse me?

Thanks.

8. ## Re: What does this notation imply?

i do not understand

9. ## Re: What does this notation imply?

Originally Posted by 99.95
I don't mean to come off as rude or ungrateful, as you are offering me help out of your spare time for nothing.
However, I get the feeling you are being vague on purpose with your replies.
If that is not the case, then I apologise, I am somewhat frustrated. If it is the case, can you please stop trying to confuse me?
Anyone of us can only help if you have a basic idea as to the material that you are asking about. Why were you asked to do a problem containing closed intervals if you do not know how they are defined?

$\displaystyle \left[ {0,1 - {n^{ - 1}}} \right] = \left\{ {x:0 \leqslant x \leqslant 1 - {n^{ - 1}}} \right\}$

$\displaystyle \\n=1,~\{0\}\\n=2,~\left[ {0, {\tfrac{1}{2}}}\right]\\n=3,~\left[ {0, {\tfrac{2}{3}}}}\right]\\\cdots$

10. ## Re: What does this notation imply?

The problem is you never did say what you did understand about the symbols and what it was that you did not. So usagi has been trying to explain every part. Did you try putting numbers into the formula? When n= 1, [0, 1- 1/n] is [0, 1-1]= [0, 0] which is the "singleton" set, {0}. When n= 2, [0, 1- 1/2]= [0, 1/2] which is the closed interval from 0 to 1/2. When n= 3, [0, 1- 1/3]= [0, 2/3] which is the closed interval from 0 to 2/3, etc.

$\displaystyle \cup_{n=2}^\infty [0, 1- \frac{1}{n}]$
means $\displaystyle [0, 0]\cup [0, 1- \frac{1}{2}]\cup [0, 1- \frac{1}{3}]\cup [0, 1-\frac{1}{4}]\cdot\cdot\cdot$
which is the same as $\displaystyle {0}\cup [0, 1/2]\cup [0, 2/3]\cup [0, 3/4]\cdot\cdot\cdot$

You should be able to see that the intervals always have left endpoint 0 and then go up to some 1- 1/n= n/n- 1/n= (n- 1)/n. As n gets larger and larger that goes to 1 (1/2, 2/3, 3/4, 4/5, ... goes to 1) but is never actually equal to 1 so we have every number from 0 (including 0) up to 1 (but NOT including 1).
$\displaystyle \cup_{n=1}^\infty [0, 1- \frac{1}{n}]= [0, 1)$
(Notice that, although every interval in the union is closed the union itself is not.)

11. ## Re: What does this notation imply?

Originally Posted by Plato
Anyone of us can only help if you have a basic idea as to the material that you are asking about. Why were you asked to do a problem containing closed intervals if you do not know how they are defined?

$\displaystyle \left[ {0,1 - {n^{ - 1}}} \right] = \left\{ {x:0 \leqslant x \leqslant 1 - {n^{ - 1}}} \right\}$

$\displaystyle \\n=1,~\{0\}\\n=2,~\left[ {0, {\tfrac{1}{2}}}\right]\\n=3,~\left[ {0, {\tfrac{2}{3}}}}\right]\\\cdots$