I am by trade a mechanical engineer, not explicitly trained in set builder notation. With that said, I am trying to learn to communicate information succinctly. What I am trying to express via set builder notation is the following:
I have a set of $m$ objects, each having their own $\alpha$ value. A set $x$ is made up of the series of the objects' $\alpha$'s, where all $\alpha$'s are confined to the real values between 0 and 1 (inclusive). My initial, seemingly verbose attempt is below, where $j$ is the object index that I use elsewhere in my document.
$$x = \{\alpha_j | \alpha_j\in[0,1]\wedge j \in \mathbb{Z} : j\in [1,m] \}$$
If you can think of a clearer way to express $x$ more cleanly, please do, and explain to me (if you have the time) your opinion/reasoning. Thank you.
Me, I'd be content with $$ x = \{a_i\}_{i=1}^{n}, \text{ where }0 \le a_i \le 1 \,\,\,\, \forall i$$
which is a pretty common notation.
Of course, this doesn't provide you with the notation to refer to the objects themselves, like the comments by DRenfro do.
Additionally, you use both the terms "set" and "series", but they are different things. Series have an underlying order, can have duplicated elements, and are usually written with parentheses. Sets are unordered, don't have duplicated elements, and use curly braces.