Am I using these intervals and symbols correctly?

Question

$a(x)$ is a function that returns single values. If $a(1)=4$ then $a_1=4$.

$S(x)$ is a function that returns sets of values. If $S(1)=\{ 4,5,6\}$ then $S_1=\{ 4,5,6\}$.

$f(x)$ is a function that returns single values. If $f(1)=4$ then $f_1=4$.

$N=\{ 1,2,3\}$ and $I=[1,3]$. (I chose $I$ for "interval.")

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These I know to be true:

$$a_1+a_2+a_3=\sum_{n=1}^{3}a_n=\sum_{n\in N}a_n$$

$$a_1\cdot a_2\cdot a_3=\prod_{n=1}^{3}a_n=\prod_{n\in N}a_n$$

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Here's where the questions begin: which, if any, of these are true?

$$S_1 \cup S_2 \cup S_3=\bigcup_{n=1}^3 S_{n}=\bigcup_{n\in N}S_n$$

$$f_1\,dx +\cdots +f_{2.\overline{99}}\, dx+f_3\, dx=\int_{1}^{3}\, dx=\int_{I}f\, dx$$

I know writing an integral like that is really an abuse of notation, but the point is that it's infinite.

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In addition, how could I notate the addition of all the elements of $N$? My attempt would be:

$$1+2+3=\sum_{c\in N}c$$

(I used $c$ to avoid confusion by using both cases of the same letter.)

Answer 1

$S_1 \cup S_2 \cup S_3=\bigcup_{n=1}^3 S_{n}=\bigcup_{n\in N}S_n$

This is fine.

$f_1\,dx +\cdots +f_{2.\overline{99}}\, dx+f_3\, dx=\int_{1}^{3}\, dx=\int_{I}f\, dx$

This is kind of right, but also terrible notation. And I assume the middle part is meant to be $\int_1^3 f(x) dx$. While an integral is (in Riemann form, at least) the limit of a sum of rectangular areas, you'd never actually write the limit itself as the sum of a bunch of infinitesimal areas. But the last equality is basically right.

In addition, how could I notate the addition of all the elements of $N$? My attempt would be: $1+2+3=\sum_{c\in N}c$ (I used $c$ to avoid confusion by using both cases of the same letter.)

That's fine. And frankly, using upper and lower cases of the same letter together is incredibly common in many situations so don't freak out too much about doing so. For example, in the same broad context I could probably find examples of $s \in S$, $\frac{N}{n}$ and $X = x$.

Is writing a set on the bottom of $\sum$, $\bigcup$, etc. legal if the set does not have nice clean numbers that all go up by $1$ (e.g., $N=\{ 1, 1.5, 2.75\}$)?

Yes, as long as that set indexes the terms you're operating on in some fashion. If that holds, then you can basically use whatever you want as long as the notation is unambiguous. For example, suppose I've indexed the digits of the number 278348923189 as $d_1 = 2$, $d_2 = 7$, $d_3 = 8$ and so forth up to $d_12$, and I want to describe a formula which is summing the odd and even digits separately, and subtracting the latter from the former. I could write that as:

$$\sum_{i \in \{1, 2, .., 12\} \\ i \text{ odd}}{d_i} - \sum_{i \in \{1, 2, .., 12\} \\ i \text{ even}} d_i$$

and besides the crummy formatting because I'm too lazy to write good LaTeX, no-one would have a problem with that. You can even do a sum over a set that isn't made of numbers. For example, in sampling you might have a population $U$ and under some kind of sampling scheme you have the set of all possible samples of $U$ denoted as $S$. Then a given sample taken from $S$, call it $s$, is just a subset of elements of $U$ (whatever they are). Then not only is it common to takes sums over $i \in s$ (i.e. add something up over all units in the sample), but also all possible $s \in S$ (i.e. add something up over all possible samples). For example, you might write something like

$$Var(\hat{Y}_\pi) = \frac{1}{\#S} \sum_{s \in S} (\hat{Y}_{\pi s} - E(\hat{Y}_\pi))^2$$

Answer 2

Every thing is fine except: $f_1dx +....+f_{2.99}dx + f_{3}dx $. [1,3] $\ne $ {1,1.01,1.02,....2.99,3}. As [1,3] is uncountable we simply can not list them in any order. We just can't.

Now it would be okay to say $\sum_{c\in [1,0]} f_cdx $ if we had some meaning to an uncountable sum which we don't-- we simply consider a sum to add one thing after another, which just makes no sense to do on an uncountable set.

But I think I see what you want to say and is okay:

Let $U_n = \{1,1+2/n,1+4/n.....,1+2(n-1)n/n=3-2/n\}$ and we let $u_{n,k}= 1+2k/n $.

Yes, we can have two dimensional indexes.

And we can have $G_n =\{f_{u_{n,k}}\} $ and we can write $f_n (k)=f_{u_{n,k}} $

Now we want $L_n =\{[u_{n,k},u_{n,k}+2/n]\} $ and $l_{n,k}=l_{u_{n,k}}= [u_{n,k},u_{n,k}+2/n]$

So we can write $\cup_{k\in U_k}l_{n,k}=\cup_{l\in L_n}l=[1,3] $

And we can write:

$F_n= f_1*l_ {n,1}+f_{1+2/n}*l_{n,2}+..... = \sum_{u\in U_n}f_u*l_u $

Then we define $\lim_{n\rightarrow \infty}F_n =\int_1^3 f (x)dx $ where $f (x)=\lim f_n (u_{n,k}) $ where $u_{n,k}$ is closest value to $x $. (I.e where $u_{n,k} = \frac 2n*floor(x*n/2))$.

Okay, that's a little overwhelming but ...

Yes, you can index $c\in A $ and you have a good grip on it.