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!
On
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 $$\frac{diam([3,4])}{diam([1,5])}$$
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?