description/proof of that cardinality of multiple times product 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 product set.
- The reader knows a definition of cardinality of set.
- The reader admits the proposition that for the cardinality of any set, any 'positive natural number'-th power of the cardinality is the natural number times multiplication of the cardinality.
Target Context
- The reader will have a description and a proof of the proposition that for any set and any positive natural number, the cardinality of the number times product 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: Structured Description
Here is the rules of Structured Description.
Entities:
\(S\): \(\in \{\text{ the sets }\}\)
\(n\): \(\in \mathbb{N} \setminus \{0\}\)
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Statements:
\(Card (S \times ... \times S) = Card S ... Card S\), where the left hand side is \(n\) times product and the right hand side is \(n\) times multiplication
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2: Note
By the proposition that for the cardinality of any set, any 'positive 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 \(Card (S)\).
3: Proof
Whole Strategy: prove it inductively; Step 1: see that it holds for \(n = 1\); Step 2: suppose that it holds for \(n = n'\) where \(1 \le n'\), and see that it holds for \(n = n' + 1\); Step 3: conclude the proposition.
Step 1:
For \(n = 1\), \(Card (S) = Card (S)\).
Step 2:
Let us suppose that for \(n = n'\) where \(n' \in \mathbb{N} \setminus \{0\}\) such that \(1 \le n'\) is any, \(Card (S \times ... \times S) = Card S ... Card S\), where the left hand side is \(n\) times product and the right hand side is \(n\) times multiplication.
\(Card (S \times ... \times S \times S)\), which is \(n + 1\) times product, is \(Card ((S \times ... \times S) \times S) = Card (S \times ... \times S) Card (S)\), by the definition of arithmetic of cardinalities.
\(= Card (S) ... Card (S) Card (S)\), by the induction hypothesis.
So, \(Card (S \times ... \times S \times S) = Card (S) ... Card (S) Card (S)\), where the left hand side is \(n + 1\) times product and the right hand side is \(n + 1\) times multiplication.
Step 3:
So, by the induction principle, for any natural number, \(n\), \(Card (S \times ... \times S) = Card S ... Card S\), where the left hand side is \(n\) times product and the right hand side is \(n\) times multiplication.
