ThmDex – An index of mathematical definitions, results, and conjectures.
Cardinality of a finite set raised to a finite power
Formulation 0
Let $X$ be a D17: Finite set.
Let $N \in \mathbb{N}$ be a D996: Natural number.
Then \begin{equation} \left| \prod_{n = 1}^N X \right| = |X|^N \end{equation}
Formulation 1
Let $X$ be a D17: Finite set.
Let $N \in \mathbb{N}$ be a D996: Natural number.
Then \begin{equation} \left| X^N \right| = |X|^N \end{equation}
Proofs
Proof 0
Let $X$ be a D17: Finite set.
Let $N \in \mathbb{N}$ be a D996: Natural number.
This result is a particular case of R1832: Cardinality of a finite cartesian product of finite sets. $\square$