767: Slicing Map on Euclidean Set

<The previous article in this series | The table of contents of this series | The next article in this series>

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:
${\mathbb{R}}^{d^{\prime }}$: $=\text{ the Euclidean set }$
$J$: $\subseteq \{1,...,d^{\prime }\}$
$r^{\prime }$: $\in {\mathbb{R}}^{d^{\prime }}$
$\ast {\lambda }_{J,r^{\prime }}$: $:Pow({\mathbb{R}}^{d^{\prime }})\to Pow({\mathbb{R}}^{d^{\prime }}),S\mapsto \{s\in S|\mathrm{\forall }j\in \{1,...,d^{\prime }\}\setminus J(s^j=r^{\prime j})\}$
//

Conditions:
//

---

2: Natural Language Description

For the Euclidean set, ${\mathbb{R}}^{d^{\prime }}$, any subset, $J\subseteq \{1,...,d^{\prime }\}$, and any point, $r^{\prime }\in {\mathbb{R}}^{d^{\prime }}$, the map, ${\lambda }_{J,r^{\prime }}:Pow({\mathbb{R}}^{d^{\prime }})\to Pow({\mathbb{R}}^{d^{\prime }}),S\mapsto \{s\in S|\mathrm{\forall }j\in \{1,...,d^{\prime }\}\setminus J(s^j=r^{\prime j})\}$

---

3: Note

The reason why we do as $\mathrm{\forall }j\in \{1,...,d^{\prime }\}\setminus J(s^j=r^{\prime j})$ instead of $\mathrm{\forall }j\in J(s^j=r^{\prime j})$ is that the remained $J$ components are usually more important than the fixed $\{1,...,d^{\prime }\}\setminus J$ components. In fact, in many cases, we do the projection that takes the $J$ components after the slicing map, getting the subset of ${\mathbb{R}}^{|J|}$.

We are saying "Euclidean set", because this concept does not require any topology, any vectors space structure, or something, although ${\mathbb{R}}^{d^{\prime }}$ is typically a Euclidean topological space or something.

---

References

<The previous article in this series | The table of contents of this series | The next article in this series>