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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.

kipf
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ZaneMorris
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    Trying to do what you're doing is probably going to introduce ambiguities (for example, one has to do some interpretation to determine how to parenthesize $\in$ followed by $\wedge$ followed by $\in$ followed by $:$ followed by $\in)$ and render the result difficult for non-machine humans to read. I think what you have could be described as follows (but this is somewhat of a guess as to what you intend): "We have objects $E_1, , E_2, , \ldots, , E_j,, \ldots, , E_m$ and a function $\alpha$ such that for each $j$ we have $\alpha(E_j) = \alpha_j \in [0,1].$" (continued) – Dave L. Renfro Aug 24 '23 at 16:17
  • However, depending on context and intended readers, it might be best to omit "and a function $\alpha$". Also, depending on context and such, some of this stuff might be front-loaded in the beginning somewhere, such as $j$ represents a positive integer between $1$ and $m,$ inclusive and the fact that for each $j$ we have $\alpha_j \in [0,1].$ – Dave L. Renfro Aug 24 '23 at 16:17

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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.

JonathanZ
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