definition of product map
Topics
About: set
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
Starting Context
- The reader knows a definition of map.
- The reader knows a definition of product set.
Target Context
- The reader will have a definition of product map.
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 1
Here is the rules of Structured Description.
Entities:
\( A\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\( \{S_\alpha\}\): \(\alpha \in A\), \(S_\alpha \in \{\text{ the sets }\}\)
\( \{S'_\alpha\}\): \(\alpha \in A\), \(S'_\alpha \in \{\text{ the sets }\}\)
\( \{f_\alpha\}\): \(\alpha \in A\), \(: S_\alpha \to S'_\alpha\)
\(*\times_{\alpha \in A} f_\alpha\): \(:\times_{\alpha \in A} S_\alpha \to \times_{\alpha \in A} S'_\alpha, (\alpha \mapsto f (\alpha)) \mapsto (\alpha \mapsto f_\alpha (f (\alpha)))\)
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Conditions:
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2: Natural Language Description 1
For any possibly uncountable index set, \(A\), any sets, \(\{S_\alpha \vert \alpha \in A\}\), any sets, \(\{S'_\alpha \vert \alpha \in A\}\), and any maps, \(\{f_\alpha: S_\alpha \to S'_\alpha\}\), \(\times_{\alpha \in A} f_\alpha :\times_{\alpha \in A} S_\alpha \to \times_{\alpha \in A} S'_\alpha\), \((\alpha \mapsto f (\alpha)) \mapsto (f': \alpha \mapsto f_\alpha (f (\alpha)))\)
3: Structured Description 2
Here is the rules of Structured Description.
Entities:
\( J\): \(= \{1, ..., n\}\)
\( \{S_j\}\): \(j \in J\), \(S_j \in \{\text{ the sets }\}\)
\( \{S'_j\}\): \(j \in J\), \(S'_j \in \{\text{ the sets }\}\)
\( \{f_j\}\): \(j \in J\), \(: S_j \to S'_j\)
\(*f_1 \times f_2 \times ... \times f_n\): \(: S_1 \times S_2 \times ... \times S_n \to S'_1 \times S'_2 \times ... \times S'_n, (p_1, p_2, ..., p_n) \mapsto (f_1 (p_1), f_2 (p_2), ..., f_n (p_n))\)
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, \(S'_1, S'_2, ..., S'_n\), and any same number of maps, \(f_1: S_1 \to S'_1, f_2: S_2 \to S'_2, ..., f_n: S_n \to S'_n\), \(f_1 \times f_2 \times ... \times f_n: S_1 \times S_2 \times ... \times S_n \to S'_1 \times S'_2 \times ... \times S'_n\), \((p_1, p_2, ..., p_n) \mapsto (f_1 (p_1), f_2 (p_2), ..., f_n (p_n))\)
