definition of disjointed union of indexed set of sets
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of indexed set.
- The reader knows a definition of product set.
Target Context
- The reader will have a definition of disjointed union of indexed set of sets.
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:
\( J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\( \{L_j \in \{\text{ the possibly uncountable sets }\}\}_{j \in J}\): \(\in \{\text{ the indexed sets }\}\)
\(*\cup_{j \in J} \{j\} \times L_j\):
//
Conditions:
//
2: Note
\(\{L_j \in \{\text{ the possibly uncountable sets }\}\}_{j \in J}\) is not disjoint in general.
So, taking \(\cup_{j \in J} L_j\) could eliminate some duplications: when \(l \in L_j \cap L_l\), \(l\) appears only once in \(\cup_{j \in J} L_j\).
For example, for \(J = \{0, 1\}\) and \(L_j = \mathbb{N}\) for each \(j \in J\), \(\cup_{j \in J} L_j = \mathbb{N}\).
So, when we need a union without any duplication eliminated, we can take the disjointed union of \(\{L_j\}_{j \in J}\), \(\cup_{j \in J} \{j\} \times L_j\).
