Let $X$ and $Y$ each be a D17: Finite set such that

(i) | $Y^X$ is the D68: Set of maps from $X$ to $Y$ |

Then
\begin{equation}
|Y^X| = |Y|^{|X|}
\end{equation}

Result R1856
on D15: Set cardinality

*Subresult of R1832: Cardinality of a finite cartesian product of finite sets*

Cardinality of the set of maps between finite sets

Formulation 0

Let $X$ and $Y$ each be a D17: Finite set such that

(i) | $Y^X$ is the D68: Set of maps from $X$ to $Y$ |

Then
\begin{equation}
|Y^X| = |Y|^{|X|}
\end{equation}

Subresults

▶ | R4314: Number of boolean functions on a finite set |

▶ | R5093: Total number of fixed-length sequences using a given number of labels |

Proofs

Let $X$ and $Y$ each be a D17: Finite set such that

(i) | $Y^X$ is the D68: Set of maps from $X$ to $Y$ |

By definition, $Y^X = \prod_{x \in X} Y$. Thus, R1832: Cardinality of a finite cartesian product of finite sets shows that
\begin{equation}
|Y^X|
= \left| \prod_{x \in X} Y \right|
= \prod_{x \in X} |Y|
= |Y|^{|X|}
\end{equation}
$\square$