a) Show that by \left(a_{1}, \ldots, a_{n}\right) \sim\left(b_{1}, \ldots, b_{n}\right): \Longleftrightarrow \exists \sigma \in S_{n}:\left(a_{1}, \ldots, a_{n}\right)=\left(b_{\sigma(1)}, \ldots, b_{\sigma(n)}\right) \quad\left(a, b \in A^{n}\right) \text { An equivalence relation of } ~ A^{n} \text { is defined. } ) Show that by f: A^{n} / \sim \rightarrow \mathbb{N}_{0}^{A},\left(a_{1, \ldots,}, a_{n}\right) \mapsto\left(\begin{array}{l}

A \rightarrow \mathbb{N}_{0} \\

a \mapsto \#\left\{i \in\{1, \ldots, n\} \mid a_{i}=a\right\}

\end{array}\right) A map is defined. c) For which sets A is f injective. d) For which sets A is f surjective. e) For which sets A is f bijective.te