A description/proof of that cardinality of multiple times multiplication of set is that times multiplication of cardinality of set
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 multiplication of set.
- The reader knows a definition of cardinality of set.
Target Context
- The reader will have a description and a proof of the proposition that for any set and any natural number, the cardinality of the number times multiplication of the set is the number times multiplication of the cardinality of the set.
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 set, \(S\), and any natural number, \(n\), \(card (S \times S \times . . . \times S) = card S card S . . . card S\), where the both hand sides multiplications are \(n\) times multiplications.
2: Proof
For \(n = 1\), \(card S = card S\).
Let us suppose that for an \(n\), \(card (S \times S \times . . . \times S) = card S card S . . . card S\) where the both hand sides multiplications are \(n\) times multiplications. \(card (S \times S \times . . . \times S)\), where the multiplication is the \(n + 1\) times multiplication, is \(card ((S \times S \times . . . \times S) \times S) = card (S \times S \times . . . \times S) card S\) by the definition of arithmetic of cardinalities. \(card (S \times S \times . . . \times S) card S = card S card S . . . card S card S\) by the supposition, so, \(card (S \times S \times . . . \times S) = card S card S . . . card S\) where the both hand sides multiplications are \(n + 1\) times multiplications.
So, by the mathematical induction principle, for any natural number, \(n\), \(card (S \times S \times . . . \times S) = card S card S . . . card S\) where the both hand sides multiplications are \(n\) times multiplications.
3: Note
By the proposition that for the cardinality of any set, any 'natural number'-th power of the cardinality is the natural number times multiplication of the cardinality, \(card (S \times S \times . . . \times S) = (card S)^n\) holds, whose right hand side is not really defined to be the \(n\) times multiplication of \(card S\).
