253: Multiplications of Cardinalities of Sets Are Associative

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A description/proof of that multiplications of cardinalities of sets are associative

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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 cardinality of set.
• The reader admits the proposition that the nested product of any sets are associative in the 'sets - map morphisms' isomorphism sense.

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Target Context

• The reader will have a description and a proof of the proposition that the nested product of any cardinalities of sets are associative.

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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 of the cardinalities, $(card{S}_{1}card{S}_2)card{S}_3$, equals $card{S}_1(card{S}_{2}card{S}_3)$.

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2: Proof

$(card{S}_{1}card{S}_2)card{S}_3=card(S_1\times S_2)card{S}_3=card((S_1\times S_2)\times S_3)$ by the definition of arithmetic of cardinalities. $card{S}_1(card{S}_{2}card{S}_3)=card{S}_{1}card(S_2\times S_3)=card(S_1\times (S_2\times S_3))$. By the proposition that the nested product of any sets are associative in the 'sets - map morphisms' isomorphism sense, $(S_1\times S_2)\times S_3)$ and $S_1\times (S_2\times S_3)$ are 'sets - map morphisms' isomorphic to each other, so, the cardinalities are the same.

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3: Note

Because of this proposition, the expression, $card{S}_{1}card{S}_{2}card{S}_3$ is possible without any ambiguity.

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References

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