Proof
(related to Proposition: Fundamental Counting Principle)
- From the rules of finite cardinals and the Cartesian product it follows for any finite sets \(X\) and \(U\) \[|X\times U|=|X|\cdot |U|,~~~~~~~~~~~~~~~~~~~~( * )\]
- By hypothesis \(S_1,\ldots S_n\) are finite sets with the cardinalities \(|S_i|\). It follows from the rule \(( * )\) that
\[\begin{array}{ccl}
|S|&=&|S_1\times\ldots\times S_{n-1}|\cdot|S_n|\\
&=&||S_1\times\ldots\times S_{n-2}|\cdot|S_{n-1}|\cdot|S_n|\\
&\vdots&\\
&=&|S_1|\cdot|S_2|\cdot\ldots\cdot|S_{n-1}|\cdot|S_n|\\
&=&\prod_{i=1}^n |S_i|
\end{array}\]
- Hereby, the associativity of Cartesian product and the associativity of the multiplication of natural numbers were used.
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References
Bibliography
- Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013
- Aigner, Martin: "Diskrete Mathematik", vieweg studium, 1993
- Ebbinghaus, H.-D.: "Einführung in die Mengenlehre", BI Wisschenschaftsverlag, 1994, 3th Edition
