Proper way to express 0 in this case?

Question

If 0=(x-a)(x-b)(x-c)...(x-x)..=0. So it's a product sum that we write with pi instead of sigma but how? There should be indexes but I'm not convinced that I understand what notation to use.

$$\prod_{ z=a }^{ y }{\left(x-z\right)} = 0$$

I don't think the expression above is correct but it's my attempt to illustrate the 0. I'm not trying to solve a specific problem. I want to learn how to formalize it if you can tell how.

Answer 1

Seems like you're using $x$ in multiple ways. Integer subscripts are a wonderful way of avoiding this.

I think the statement you want is $0 = (x-a_1) (x- a_2) \ldots (x- a_n)$, which can also be written as $\prod_{i=1}^n (x-a_i) = 0$.

Answer 2

You could try defining a family like: $$ \mathcal F = \{a, b, c, \ldots, x, \ldots, z\} $$ and then doing: $$ \prod_{\alpha \in \mathcal F} (x - \alpha) = 0 $$