definition of cross section of subset of product set by element of 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 cross section of subset of product set by element of 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:
\( J'\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\( \{S_{j'} \in \{\text{ the sets }\} \vert j' \in J'\}\):
\( \times_{j' \in J'} S_{j'}\): \(= \text{ the product set }\)
\( S\): \(\subseteq \times_{j' \in J'} S_{j'}\)
\( J\): \(\subset J'\), such that \(J \neq \emptyset\)
\( \times_{j \in J} s_j\): \(\in \times_{j \in J} S_j\)
\(*S_{[\times_{j \in J} s_j]}\): \(= \{\times_{l \in J' \setminus J} s_l \in \times_{l \in J' \setminus J} S_l \vert \times_{j' \in J'} s_{j'} \in S\}\), \(\subseteq \times_{l \in J' \setminus J} S_l\)
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Conditions:
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2: Note
Narrowly speaking, "cross section" may require that \(\vert J \vert = 1\), but this definition is a generalization of that narrow "cross section".
\(S_{[\times_{j \in J} s_j]}\) may be empty.
