A description/proof of that multiplications of cardinalities of sets are associative
Topics
About: set
The table of contents of this article
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.
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.
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: 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 \(card S_1 (card S_2 card S_3)\).
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.
3: Note
Because of this proposition, the expression, \(card S_1 card S_2 card S_3\) is possible without any ambiguity.
