What is an Index Set and How is it Defined?

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    Definition Index Set
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Discussion Overview

The discussion revolves around the concept of an index set, seeking a precise definition and understanding of its role in mathematical notation and operations. Participants explore its usage in contexts such as unions of sets and the necessity of defining related terms like "index."

Discussion Character

  • Exploratory, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant requests a precise definition of an index set, indicating a lack of clarity on the term.
  • Another participant defines an index set as a set used solely for indexing, providing an example with a union of sets.
  • This participant suggests that index sets are similar to integration variables, emphasizing their role in notation rather than in substantive mathematical content.
  • Some participants argue that defining "index" is necessary before defining "index set," indicating a potential foundational issue in understanding the term.
  • A later reply humorously requests definitions for both "index" and "index set," highlighting the ongoing uncertainty in the discussion.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the definition of an index set, and there is disagreement regarding the necessity of defining "index" first. Multiple views on the topic remain present.

Contextual Notes

The discussion highlights a potential limitation in the definitions provided, as the term "index" remains undefined, which may affect the understanding of "index set." There is also ambiguity in the role and significance of index sets in mathematical notation.

Buri
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Can someone give me a precise definition of what an index set is? I have never seen it defined.

Thanks!
 
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An index set is just a set which only use is to index something.
For example, in [tex]\bigcup_{i\in \{1,2\}}{A_i}[/tex], the set [tex]\{1,2\}[/tex] is an index set.

Sometimes you'll see "Take an index set I and consider the union [tex]\bigcup_{i\in I}{A_i}[/tex]..." This just means that you can take any set and consider the union. There is nothing special about I, it can be about anything.

An index set is like an integration variable, it's only use is to make your notation work. For the rest, it isn't used.
 
Before defining "index set" you will have to define "index"!
 
HallsofIvy said:
Before defining "index set" you will have to define "index"!

Could you define both then? lol
 

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