252: Products of Sets Are Associative in 'Sets - Map Morphisms' Isomorphism Sense

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A description/proof of that products of sets are associative in 'sets - map morphisms' isomorphism sense

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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: Description
• 2: Proof
• 3: Note

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Starting Context

• The reader knows a definition of set.
• The reader knows a definition of product of sets.
• The reader knows a definition of %category name% isomorphism.

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Target Context

• The reader will have a description and a proof of the proposition that the nested product of any sets are associative in the 'sets - map morphisms' isomorphism sense.

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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: Description

For any sets, $S_1,S_2,S_3$, the nested product, $(S_1\times S_2)\times S_3$ is 'sets - map morphisms' isomorphic to $S_1\times (S_2\times S_3)$.

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2: Proof

$(S_1\times S_2)\times S_3=\{s|\mathrm{\exists }{s}_1\in S_1,\mathrm{\exists }{s}_2\in S_2,\mathrm{\exists }{s}_3\in S_3,s=⟨⟨s_1,s_2⟩,s_3⟩\}$ is obviously not exactly $S_1\times (S_2\times S_3)=\{s|\mathrm{\exists }{s}_1\in S_1,\mathrm{\exists }{s}_2\in S_2,\mathrm{\exists }{s}_3\in S_3,s=⟨s_1,⟨s_2,s_3⟩⟩\}$, but $f:(S_1\times S_2)\times S_3\to S_1\times (S_2\times S_3),⟨⟨s_1,s_2⟩,s_3⟩\mapsto ⟨s_1,⟨s_2,s_3⟩⟩$ is a bijection.

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3: Note

Although sloppy expressions like "$(S_1\times S_2)\times S_3=S_1\times (S_2\times S_3)$" are prevalently seen, the 2 are not the same, but are isomorphic to each other.

The expression, $S_1\times S_2\times S_3$, is allowed because it is defined as $(S_1\times S_2)\times S_3$, not because "$(S_1\times S_2)\times S_3=S_1\times (S_2\times S_3)$" holds.

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References

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