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...
Let b be an integer greater than or equal to 2. I think I write this as: $$\{b\in\Bbb Z\,:\,b\ge 2\}$$
Let c be an integer in the range 1 to (b-1). I think I write this as: $$\{c\in\Bbb Z\,:\,0<c<b\}$$
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?
This is the set of all such numbers. You could just write: $$ b \in \mathbb{Z}, b \ge 2. $$ or more like your version $$ b \in \{ b' \in \mathbb{Z} \mid b' \ge 2 \} $$
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 $$ c \in \{ c' \in\mathbb{Z} \mid 0 < c'< b \} $$
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}