ThmDex – An index of mathematical definitions, results, and conjectures.

ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4315 on D326: Cartesian product

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. $\square$