Mapping/Take preimage/Complete assignment/Exercise

Let ${}L$ and ${}M$ denote sets. We consider the mapping

$\Psi \colon \operatorname {Map} \,{\left(L,M\right)}\longrightarrow \operatorname {Map} \,{\left({\mathfrak {P}}\,(M),{\mathfrak {P}}\,(L)\right)},f\longmapsto f^{-1},$

which assigns to a mapping the mapping given by taking preimages.

a) Show that ${}\Psi$ is injective.

b) Suppose ${}L\neq \emptyset$. Show that ${}\Psi$ is not surjective.

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