# How to write equation for a set of products with specific range limits?

I hope you welcome non-mathematicians here. I am trying to find out how to write the following idea as a math equation, currently it is in computer code but I want to be able to discuss this with some of the math guys at my work and at least start the conversation in their terminology. I will show what I think I know and I will hope to learn from everyone here. Thank you in advance.

I will start by defining the variables involved...

1. Let b be an integer greater than or equal to 2. I think I write this as: $$\{b\in\Bbb Z\,:\,b\ge 2\}$$

2. Let c be an integer in the range 1 to (b-1). I think I write this as: $$\{c\in\Bbb Z\,:\,0<c<b\}$$

3. Finally there is a function of c that will resolve as a real number. I think I write this as: $$f(c) = \mathbb{R}$$

What I need to do is take a sequence of expressions and multiply all the members of that sequence to produce a single real number. The basic form of each expression can be simplified to: $${f(c) \over b}$$ and then for each value of b an equation using a sequence of expressions is created as follows.

For b = 2 the equation is simply: $$A = \left\{ {f(1) \over b} \right\}$$

For b = 3 the equation is: $$A = \left\{ {f(1) \over b} \times {f(2) \over b} \right\}$$

and so on.

In general the equation would look something like...

For b = n:

$$A = \left\{ {f(1) \over b} \times {f(2) \over b} \times \ldots \times {f(n-1) \over b}\right\}$$

but because of the first case (b = 2) I am not sure how to properly write this.

So, how did I do as far as figuring this out on my own? What did I get wrong and how do I fix it?

1. Let b be an integer greater than or equal to 2. I think I write this as: $$\{b\in\Bbb Z\,:\,b\ge 2\}$$

This is the set of all such numbers. You could just write: $$b \in \mathbb{Z}, b \ge 2.$$ or more like your version $$b \in \{ b' \in \mathbb{Z} \mid b' \ge 2 \}$$

1. Let c be an integer in the range 1 to (b-1). I think I write this as: $$\{c\in\Bbb Z\,:\,0<c<b\}$$

Same issue. Just write $$c \in \{1, 2, \dotsc, b-1 \}$$ or if the dots are not rigorous enough $$c \in [1, b-1] \cap \mathbb{Z}$$ or $$c \in \{ c' \in\mathbb{Z} \mid 0 < c'< b \}$$

1. Finally there is a function of c that will resolve as a real number. I think I write this as: $$f(c) = \mathbb{R}$$

If you want to give a rough description of $f$, provide its domain and codomain sets: $$f : A \to \mathbb{R}$$ or $$f \in \mathbb{R}^A$$ and decide how precise you want to describe your domain set $A$, is it just $\mathbb{N}$ or $\{ c \in \mathbb{Z} \mid 0 < c < b \}$?

If you want only to describe the codomain set of $f$ then $f(c) \in \mathbb{R}$ might be enough.

I am not sure about your sequence, it might be $$s_b = \prod_{k=2}^b \frac{f(k-1)}{b}$$ then \begin{align} s_2 &= \frac{f(1)}{2} \\ s_3 &= \frac{f(1)}{3}\frac{f(2)}{3} \\ & \vdots \\ s_n &= \frac{f(1)}{n}\frac{f(2)}{n} \dotsb \frac{f(n-1)}{n} \end{align}

• Wonderful. I will need to do some reading to understand the product notation but it seems similar to sum notation so I think I will grasp it with a little effort. Everything else I am familiar with (though a little wobbly on) from my college algebra classes. On the matter of $f(c)$ my intent here was just to describe the codomain as $\Bbb R$ for simplification of this question, with more formal descriptions being used with different functions of $c$. Thank you mvw. I will go do that reading and come back in a few days to either ask a little more or else accept the answer as given. – Mathew Ambrose May 20 '18 at 13:42
• There are different ways to express different functions depending on some parameter $c$. E.g. they might form a family of curves. One way is e.g. $f_c(x) = x^2 + c$ another $f(x;c) = x^2 + c$ or simply $f(x,c) = x^2 + 2$. But maybe different functions are very different. It depends a bit on the concrete example. – mvw May 20 '18 at 15:21
• I understand. This is part of what I want to talk to the guys at work about, figuring out the best $f(c)$ to get the desired results for the software routine I am developing. Sometimes they can be a bit snobbish because I am not a math guy so I try to show them I am making an effort to work in their world whenever I need their help. It is my way of showing respect for their expertise. – Mathew Ambrose May 20 '18 at 16:39
• Computer programs can be described at various levels, from real code to pseudo code to a very mathematical modeling in the realm of theoretical computer science. The basic motivation should be to find a common ground such that everyone involved gets it. Hard to give you a good recommendation without knowing more background. You might overdo it the way we discussed it here. :-) – mvw May 20 '18 at 16:54
• I appreciate what you are saying mvw, but it would be tricky to go into more detail without exposing some proprietary algorithms involved. That is why I dramatically simplified the sequence expression to a bare bones function and quotient. Frankly I would love to just lay it all out here and get the help of all the SE people both CS and Math. Sure would make my life a lot easier if I could. I do appreciate the help you have given and the candid advice as well. – Mathew Ambrose May 20 '18 at 20:28