6.3.19 Boolean algebra

Problem: 1. From the table for the Boolean function \( g\left(x_{1}, x_{2}, x_{3}\right) \) write down the Zhegalkin polynomial, perfect DNF and CNF. 2. Draw the Boolean function \( g\left(x_{1}, x_{2}, x_{3}\right) \) on a Boolean cube and find the minimal DNF. 3. Find the minimal DNF for function \( f\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \). 4. Examine the completeness of the system \( \{f, g\} \). If the system is not complete, add such a function \( h \), that the system \( \{f, g, h\} \) is complete. 5. Implement functions using a circuit of functional elements in the basis \( \left\{\wedge, \mathrm{v}^{-}\right\} \). 6. Using the functions of one of the bases of 4-th option, express functions \( 0,1, z, \bar{x}, x_{1} x_{2}, x_{1} \vee x_{2}, x_{1} \oplus x_{2} \). Implement them on the same basis with a circuit of functional elements. 7. Construct a contact circuit for the function \( g\left(x_{1}, x_{2}, x_{3}\right) \), presenting it in the form of DNF and CNF. \[ \begin{array}{l} f\left(x_{1}, x_{2}, x_{3}, x_{4}\right)=(0000011001000111), \\ g\left(x_{1}, x_{2} x_{3}\right)=\left(\overline{x_{1}} x_{2} \rightarrow x_{3}\right) \oplus\left(\overline{x_{3}} \leftrightarrow x_{1}\right) . \end{array} \]