definition of slicing map on Euclidean set
Topics
About: set
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of Euclidean set.
- The reader knows a definition of map.
Target Context
- The reader will have a definition of slicing map on Euclidean 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:
\( \mathbb{R}^{d'}\): \(= \text{ the Euclidean set }\)
\( J\): \(\subseteq \{1, ..., d'\}\)
\( r'\): \(\in \mathbb{R}^{d'}\)
\(*\lambda_{J, r'}\): \(: Pow (\mathbb{R}^{d'}) \to Pow (\mathbb{R}^{d'}), S \mapsto \{s \in S \vert \forall j \in \{1, ..., d'\} \setminus J (s^j = r'^j)\}\)
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Conditions:
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2: Natural Language Description
For the Euclidean set, \(\mathbb{R}^{d'}\), any subset, \(J \subseteq \{1, ..., d'\}\), and any point, \(r' \in \mathbb{R}^{d'}\), the map, \(\lambda_{J, r'}: Pow (\mathbb{R}^{d'}) \to Pow (\mathbb{R}^{d'}), S \mapsto \{s \in S \vert \forall j \in \{1, ..., d'\} \setminus J (s^j = r'^j)\}\)
3: Note
The reason why we do as \(\forall j \in \{1, ..., d'\} \setminus J (s^j = r'^j)\) instead of \(\forall j \in J (s^j = r'^j)\) is that the remained \(J\) components are usually more important than the fixed \(\{1, ..., d'\} \setminus J\) components. In fact, in many cases, we do the projection that takes the \(J\) components after the slicing map, getting the subset of \(\mathbb{R}^{\vert J \vert}\).
We are saying "Euclidean set", because this concept does not require any topology, any vectors space structure, or something, although \(\mathbb{R}^{d'}\) is typically a Euclidean topological space or something.
