definition of indexed set
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of map.
Target Context
- The reader will have a definition of indexed set.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\( S\): \(\in \{\text{ the sets }\}\)
\(*\{s_j \in S\}_{j \in J}\): \(\in \{\text{ the maps from } J \text{ into } S\}\)
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Conditions:
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2: Note
'indexed set' is really a map, \(f: J \to S\), and \(s_j\) is really \(f (j)\).
'indexed set' is different from 'set' in that any set has no duplication in the elements while \(s_j = s_l\) for some \(j \neq l\) is possible: \(f\) does not need to be injective.
A set indexed is not any indexed set: \(\{s_j \vert j \in J\}\) is a set indexed, which is different from the indexed set, \(\{s_j\}_{j \in J}\): \(\{s_j \vert j \in J\}\) is a set just that the elements happen to be distinguished by the index; for example, the elements of a countable set may be distinguished as \(\{s_j \vert j \in \mathbb{N}\} = \{s_1, s_2, ...\}\), but it is not any indexed set, while a sequence, \(\{s_j\}_{j \in \mathbb{N}}\), is really an indexed set.
Sometimes, a term like "indexed family" is used, but as far as \(\{s_j \in S\}_{j \in J}\) is a map, it is indeed as set, because any map is a set, so, the term, "set", does not need to be avoided.
As far as \(J\) is a set and there is a formula that determines \(s_j\) for each \(j\), \(S\) exists as a set by the replacement axiom, and \(\{s_j \in S\}_{j \in J}\) is indeed a map.
For any \(j \in J\), \(f (j)\) is called "element of \(\{s_j \in S\}_{j \in J}\)".
For any \(J^` \subseteq J\), \(f \vert_{J^`}\) is called "indexed subset".
While it is really just a map, why do we need to bother to concoct a new term instead of just using "map"?
Well, we do not exactly need to, but it is sometimes convenient, because the emphases are different: any map is about the rule of a mapping, while any indexed set is about the images of a mapping where the exact mapping is not really the issue: whether the elements are indexed as \(\{s_1, s_2, ...\}\) or \(\{s_0, s_1, ...\}\) is not really important although duplications need to be allowed, and it is a "duplication-allowed set" in a way.
