Consider a collection of sets $B=\{A_i: i\in \mathbb{N}\}$ and assume I want to work with some some set in this collection that satisfies property $P$. Assume also that at least one set in $B$ has property $P$.
Which of the following is appropriate?
- Choose some $A_i \in B$ such that $P$ holds.
- Choose some $A_j \in B$ such that $P$ holds.
The first option re-uses the indexing variable $i$. Is this allowed? Is this conveying that I am working with any element in the collection satisfying property $P$ without fixing which set I am working with exactly?
The second option introduces a new indexing variable $j$. Is this appropriate since it tells the reader that I am fixing a specific $j\in\mathbb{N}$ and am now working with the fixed set $A_j$. (Though all that is known about $A_j$ is that is an element of $B$ satisfying property $P$.
I have seen this done multiple ways and it is always clear what is being conveyed. My goal is to figure out what, if anything, is an abuse of notation, and what is technically correct. Also, I want to know any implications of re-using the index variable. I am looking for more than just "You interpret both statements the same" unless there is a formal reason for it (perhaps referencing the logic notion of variable substitution). Finally, if my notion of what it means to "fix" a set with a given property when multiple such sets may exist is not well-defined, please elaborate on that idea.
Preferences vary from individual to individual, hence there can be no Single Correct Answer.
Having said that , these are my thoughts on this situation :
(A) In "Choose some $A_i \in B$ with $P$" , it is reusing the Index $i$ which is correct , but we might confuse that with the earlier $i$.
(B) In "Choose some $A_j \in B$ with $P$" , it is not reusing the Index which is also correct , but we have no information about $j$ , whether it is Even or ODD or Single Digit.
Consider these 3 alternatives :
(C) "Choose some $C \in B$ with $P(C)$" where there is no reuse of Index & there is no Index at all !
Your Question : "Is this conveying that I am working with any element in the collection satisfying property P without fixing which set I am working with exactly?"
This is the Exact thing that this is conveying !
(D) "Choose some $i \in N$ such that [ $A_i \in B$ with ] $P(A_i)$" [ the Part in Brackets in redundant but adds clarity & can be removed ]
In this case, you can (re-)use which-ever Index variable you want , because you are (re-)introducing it & again saying that it is in $N$ (You can change this to Even or ODD or Single Digit suitable to use in your context).
(E) "Let $E=\{ U : (U \in B) \land P(U) \}$ , Choose some $V \in E$"
Here too , we are not concentrating on the Index & more-over we are having the Possibility that there are multiple elements with the Property $P$. We might use Set E when we want to make a general Statement about $V$ or we might Prove that Set E is a Singleton.
Thoughts about "Scope" of the Index :
In $U+\sum_{i=1}^{i=N}{(i+V)}$ , the Index is $i$ & we can use that only here , because it is in "Scope" whereas $N$ , $U$ & $V$ are not limited.
The same thing occurs in the Index Set Case, where earlier $i$ is limited to the earlier "Scope" & the next $i$ is limited to the next "Scope".
Why we should avoid that :
(1) When we re-use the Index , we are making it a little harder to see the whole , & we are introducing unnecessary (avoidable) confusion.
(2) Unlike Computer Programming , there is no Compiler error when we use out-of-scope variable or Index. We can still reach in later, to state (eg) that $i$ must be between $100$ & $1000$ , then it is ambiguous which $i$ we are talking about. When use use $i$ & $j$ , that ambiguity will not occur.
Why we should still use that :
It is a very "Common Practice" to use & reuse Index variable $i$ , while making sure that we are not mixing the various Instances or making unnecessary ambiguity.