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

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

Proof 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 $Y$

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} $\square$