Boolean algebra

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Problem: Find the cardinality of the set \[ A=(L \cap S) \backslash\left(T_{0} \cap T_{1}\right), \] where \( L, S, T_{0}, T_{1} \) are Post classes.

6.3.16 Boolean algebra

3.81 $

Problem: Find the power of the set of Boolean functions of \( n \) variables \( A=\left(T_{0} \cap T_{1}\right) \cup(L \backslash S) \), where \( L, S, T_{0}, T_{1} \) are Post classes.

6.3.17 Boolean algebra

3.81 $

Problem: Find the power of the set of Boolean functions with \( n \) variables. \( A=(L \backslash S) \cup\left(T_{0} \backslash T_{1}\right) \), where \( L, S, T_{0}, T_{1} \) are Post classes.

6.3.18 Boolean algebra

3.81 $

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. \[ \begin{array}{l} f\left(x_{1}, x_{2}, x_{3}, x_{4}\right)=(0000011001000111), \\ g\left(x_{1}, x_{2} x_{3}\right)=\left(\overline{x_{1}} x_{2} \rightarrow x_{3}\right) \oplus\left(\overline{x_{3}} \leftrightarrow x_{1}\right) . \end{array} \]

6.3.19 Boolean algebra

17.8 $

Problem: This formula can be converted into PDNF in two ways: 1 ) using the truth table, 2 ) by transformation. a) \( \bar{x} \vee z \), b) \( \overline{(x \vee y)(\bar{x} \vee z)} \).

6.3.20 Boolean algebra

2.54 $

Problem: From the formula go to the truth table and use the table to construct the perfect CNF. \[ \bar{x} \vee \overline{(y z)} x \text {. } \]

6.3.21 Boolean algebra

0 $

Problem: Convert the formula to Zhegalkin polynomial. \[ (x \overline{y z}) \mid(x \mid(y \mid z)) \text {. } \]

6.3.22 Boolean algebra

0 $

Problem: Identify dummy variables in functions, defined by formulas and express them in a formula containing only essential variables. \[ (x y \vee \bar{x} \bar{y}) \oplus \bar{x} . \]

6.3.23 Boolean algebra

1.27 $

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