Duality of Boolean functions

A Boolean function $f^{*}\left(x_{1}, \ldots , x_{n}\right)$ is called dual to the function $f\left(x_{1}, \ldots , x_{n}\right)$ if $f^{*}\left(x_{1}, \ldots , x_{n}\right)=\neg f\left(\neg x_{1}, \ldots , \neg x_{n}\right)$ for every assignment of values to the variables. For example, conjunction is dual to disjunction and vice versa.

A Boolean function $f^{*}\left(x_{1}, \ldots , x_{n}\right)$ is called dual to a function $f\left(x_{1}, \ldots , x_{n}\right)$ if $f^{*}\left(x_{1}, \ldots , x_{n}\right)=\neg f\left(\neg x_{1}, \ldots , \neg x_{n}\right)$ for every tuple of values of the variables. For example, conjunction is dual to disjunction and vice versa.

Construct the dual functions for the functions

Construct the dual functions for the functions given by the following formulas:

Prove that a Boolean function $f$ is self-dual if and only if $f^{*}=f$ (see problem 151 on p. 49).

Prove that an $n$-ary Boolean function $f\left(x_{1}, \ldots , x_{n}\right)$ is self-dual if and only if

for some $(n-1)$-ary Boolean function $g$.

Let $A$ be a formula with tight negations (see Problem II.1.22), and let $A_{1}$ be obtained from $A$ by replacing by $\vee$, $\vee$ by , and the variables $A_{i}$ by $\neg A_{i}$. Prove that $A_{1} \sim \neg A$.

Let $A$ and $B$ be formulas with tight negations (see Problem II.1.22), and let $A^{*}$, $B^{*}$ be the formulas dual to $A$ and $B$ respectively ($A^{*}$ is obtained from $A$ by replacing by $\vee$, $\vee$ by ). Prove that $A \sim B$ implies $A^{*} \sim B^{*}$ (the duality law).