91Ó°ÊÓ

. Let \(f: B^{4} \rightarrow B\). Find the disjunctive normal form for \(f\) if a) \(f^{-1}(1)=\\{0101\) (that is, \(w=0, x=1, y=0, z=1)\), \(0110,1000,1011\\} .\) b) \(f^{-1}(0)=\\{0000,0001,0010,0100,1000,1001,0110\\}\).

For \(n\) Boolean variables there are \(2^{2^{n}}\) Boolean functions, each of which can be represented by a function table. a) A Boolean function \(f\) on the \(n\) variables \(x_{1}, x_{2}, \ldots, x_{n}\) is called self-dual if $$ f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\overline{f\left(\bar{x}_{1}, \bar{x}_{2}, \ldots, \bar{x}_{n}\right)} $$ How many Boolean functions on \(n\) variables are self-dual? b) Let \(f: B^{3} \rightarrow B\). Then \(f\) is called symmetric if $$ \begin{aligned} f(x, y, z) &=f(x, z, y)=f(y, x, z) \\ &=f(y, z, x)=f(z, x, y)=f(z, y, x) \end{aligned} $$ So the value of \(f\) is unchanged when we rearrange the three columns of values listed under \(x, y\), and \(z\) in the table for \(f\). How many such functions are there on three Boolean variables? How many are there on \(n\) Boolean variables?