436: Product Set

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A definition of product set

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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: Definition 1
• 2: Definition 2
• 3: Note

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Starting Context

• The reader knows a definition of function.
• The reader knows a definition of tuple.

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Target Context

• The reader will have a definition of product set.

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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: Definition 1

For any possibly uncountable indices set, $A$, and any sets, $\{S_{\alpha }|\alpha \in A\}$, indexed with $A$, the set of all the functions, $\{f:A\to {\cup }_{\alpha \in A}{S}_{\alpha }|f(\alpha )\in S_{\alpha }\text{ for each }\alpha \in A\}$, denoted as ${\times }_{\alpha \in A}{S}_{\alpha }$

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2: Definition 2

For any finite number of sets, $S_1,S_2,...,S_n$, the set of all the $n$-tuples, $\{⟨p_1,p_2,...,p_n⟩|p_j\in S_j\text{ for each }j=1\sim n\}$, denoted as $S_1\times S_2\times ...\times S_n$

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3: Note

Definition 1 is indeed a set: the set of all the functions from $A$ into ${\cup }_{\alpha \in A}{S}_{\alpha }$ is a set (see Proof 8 of the proposition that some expressions can be parts of legitimate formulas for the ZFC set theory), and the formula for the subset axiom is legitimate.

Definition 2 is indeed a set, by the proposition that the product of any finite number of sets is a set.

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References

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