I'm aware that a set of indexed items $X = \{x_1,\dots,x_n\}$ is really a mapping $x: I \to X$, where $x(i)$ is denoted $x_i$ (per this question), but my question relates to mathematical writing using the heuristic notation $\{x_1,\dots,x_n\}$
Suppose I have a set of indexed items $X := \{x_1,\dots,x_n\}$, and another set of generic items $A$. There seems to be a difference between the meaning of the $\in$ in the following two statements:
- For all $x_i \in X$ (some property related to $x_i$...)
- For all $x \in A$ (some property related to $x$...)
In the first case, I have already defined $x_i$ to be a member of $X$ and is an already globally bound variable so that the inclusion sign $\in$ is merely reminding the reader that it belongs in set $X$. The property related to $x_i$ will relate to $i$. In the second case I am binding the previously free variable $x$ to range over the universe $A$, and $x$ is a generic, undistinguished member.
Based on this difference I am wondering which of the two notations is more standard/more clear. If I need to define a symbol $L$ indexed by each $x_i$, is it preferable to write:
- Let $L_{x_i}$ $(x_i \in X)$ be the (...), or
- Let $L_{x_i}$ $(i=1,2,\dots,n)$ be the (... )?
Later I will need to use the notation $L_x$ when I do not know the particular $i$ corresponding to the $x \in X$, so it is important that $L$ is indexed generically by items in $X$, and not $i$.