definition of projection from product set onto subproduct set
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of product set.
Target Context
- The reader will have a definition of projection from product set onto subproduct 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:
\( J'\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\( \{S_{j'} \in \{\text{ the sets }\} \vert j' \in J'\}\):
\( \times_{j' \in J'} S_{j'}\): \(= \text{ the product set }\)
\( J\): \(\subseteq J'\), such that \(J \neq \emptyset\)
\( \times_{j \in J} S_j\): \(= \text{ the product set }\)
\(*\pi^{J}\): \(: \times_{j' \in J'} S_{j'} \to \times_{j \in J} S_j, \times_{j' \in J'} s_{j'} \mapsto \times_{j \in J} s_j\)
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Conditions:
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2: Note
Narrowly speaking, "projection" may require that \(\vert J \vert = 1\), but this definition is a generalization of that narrow "projection".
But \(\vert J \vert = 1\) is the most prevalent case, and often, "projection" means that \(\vert J \vert = 1\).
