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Is there any way to write an equation that means the proportion between two intervals?

For example, if I have the intervals [3,4] and [1,5] I would like to know the proportion between them, but I'm not sure if $\dfrac{[3,4]}{[1,5]}$ would mean that. Is there any to calculate the percentage of the coverage of the first interval over the second?

Or... is there any notation that means the size of the interval (something analogue to the |Z| that represents the size of a set)?

Thanks!

user136913
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2 Answers2

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The size (or length) of the first interval is $4-3=1$, and the length of the second interval is $5-1=4$. Your fraction therefore is 1/4.

In general you can calculate the length using the absolute value function. $|4-3| = |3-4| = 1$.

You have to talk about the length of the intervals, since there are infinitely many real numbers in any interval, so you can't talk about the size of them as if they are sets.

Is that what you mean?

naslundx
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  • Yes the result is 1/4, but there is any form to write it in an equation using both intervals? For example: with sets I could use $\dfrac{|Z|}{|Q|}$, and with intervals? $\dfrac{|[3,4]|}{|[1,5]|}$? – user136913 Mar 26 '14 at 22:18
  • @user136913 You would have to write it as $\frac{|3-4|}{|1-5|} = \frac{1}{4}$. Your intervals are sets, but using the set-theoretical clammers as you do would be incorrect here. – naslundx Mar 26 '14 at 22:21
  • The problem is that I need something more "generic", I tought of this possibility $\dfrac{max([a,b])-min([a,b])}{max([a,b])-min([a,b])}$, but it seems to much for a simple equation :/. – user136913 Mar 26 '14 at 22:28
  • How and why do you need it? You can of course invent any kind of notation, but that does not mean the calculations become any easier. – naslundx Mar 26 '14 at 22:29
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Let $d(x,y)=|x-y|$ be the usual distance on $\mathbb{R}$ (it corresponds to the standard topology). Then what you're seeking is $$\frac{d(3,4)}{d(1,5)}=\frac{|3-4|}{|1-5|}=\frac{1}{4}$$ In the same spirit, we can define the diameter of a real set $A$ $$diam(A)=\{d(x,y);~~x,y\in A\}$$ Then we can write the ratio as $$\frac{diam([3,4])}{diam([1,5])}$$

user5402
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