524: Product Map

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definition of product map

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Topics

About: set

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description 1
• 2: Natural Language Description 1
• 3: Structured Description 2
• 4: Natural Language Description 2

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Starting Context

• The reader knows a definition of map.
• The reader knows a definition of product set.

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Target Context

• The reader will have a definition of product map.

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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.

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Main Body

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1: Structured Description 1

Here is the rules of Structured Description.

Entities:
$A$: $\in \{\text{ the possibly uncountable index sets }\}$
$\{S_{\alpha }\}$: $\alpha \in A$, $S_{\alpha }\in \{\text{ the sets }\}$
$\{S_{\alpha }^{\prime }\}$: $\alpha \in A$, $S_{\alpha }^{\prime }\in \{\text{ the sets }\}$
$\{f_{\alpha }\}$: $\alpha \in A$, $:S_{\alpha }\to S_{\alpha }^{\prime }$
$\ast {\times }_{\alpha \in A}{f}_{\alpha }$: $:{\times }_{\alpha \in A}{S}_{\alpha }\to {\times }_{\alpha \in A}{S}_{\alpha }^{\prime },(\alpha \mapsto f(\alpha ))\mapsto (\alpha \mapsto f_{\alpha }(f(\alpha )))$
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Conditions:
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2: Natural Language Description 1

For any possibly uncountable index set, $A$, any sets, $\{S_{\alpha }|\alpha \in A\}$, any sets, $\{S_{\alpha }^{\prime }|\alpha \in A\}$, and any maps, $\{f_{\alpha }:S_{\alpha }\to S_{\alpha }^{\prime }\}$, ${\times }_{\alpha \in A}{f}_{\alpha }:{\times }_{\alpha \in A}{S}_{\alpha }\to {\times }_{\alpha \in A}{S}_{\alpha }^{\prime }$, $(\alpha \mapsto f(\alpha ))\mapsto (f^{\prime }:\alpha \mapsto f_{\alpha }(f(\alpha )))$

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3: Structured Description 2

Here is the rules of Structured Description.

Entities:
$J$: $=\{1,...,n\}$
$\{S_j\}$: $j\in J$, $S_j\in \{\text{ the sets }\}$
$\{S_j^{\prime }\}$: $j\in J$, $S_j^{\prime }\in \{\text{ the sets }\}$
$\{f_j\}$: $j\in J$, $:S_j\to S_j^{\prime }$
$\ast f_1\times f_2\times ...\times f_n$: $:S_1\times S_2\times ...\times S_n\to S_1^{\prime }\times S_2^{\prime }\times ...\times S_n^{\prime },(p_1,p_2,...,p_n)\mapsto (f_1(p_1),f_2(p_2),...,f_n(p_n))$


Conditions:
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4: Natural Language Description 2

For any finite number of sets, $S_1,S_2,...,S_n$, any same number of sets, $S_1^{\prime },S_2^{\prime },...,S_n^{\prime }$, and any same number of maps, $f_1:S_1\to S_1^{\prime },f_2:S_2\to S_2^{\prime },...,f_n:S_n\to S_n^{\prime }$, $f_1\times f_2\times ...\times f_n:S_1\times S_2\times ...\times S_n\to S_1^{\prime }\times S_2^{\prime }\times ...\times S_n^{\prime }$, $(p_1,p_2,...,p_n)\mapsto (f_1(p_1),f_2(p_2),...,f_n(p_n))$

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References

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