Boolean algebra

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Problem: Prove the identity, using the laws of Boolean algebra. Represent one of the expressions in the basis of elementary functions: \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|} \hline\( y_{1} \) & \( y_{2} \) & \( y_{3} \) & \( y_{4} \) & \( y_{5} \) & \( y_{6} \) & \( y_{7} \) & \( y_{8} \) & \( y_{9} \) & \( y_{10} \) & \( y_{11} \) \\ \hline 0 & \begin{tabular}{l} \( a \) \\ \( \wedge b \) \end{tabular} & \( a \) & \( a \oplus b \) & \begin{tabular}{l} \( a \) \\ \( \vee \vee b \) \end{tabular} & \begin{tabular}{l} \( a \) \\ \( \downarrow b \) \end{tabular} & \begin{tabular}{l} \( a \) \\ \( \Leftrightarrow b \) \end{tabular} & \( \bar{a} \) & \begin{tabular}{l} \( a \) \\ \( \rightarrow b \) \end{tabular} & \( a \mid b \) & 1 \\ \hline \end{tabular} The set of basic function numbers should include the numbers of your option. For example, for option 1 \( y_{1}, y_{11}, y_{10} \) can be taken, the missing functions are selected based on Post's completeness theorem. \[ \begin{array}{l} ((a \wedge \bar{c}) \downarrow(b \wedge \bar{c})) \wedge((a \mid d)(\overrightarrow{b \wedge d}))= \\ =((a \mid b) \mid(a \oplus \bar{b})) \rightarrow((c \oplus d) \wedge(d \rightarrow c)), \text { option } 2 . \end{array} \]

6.3.1 Boolean algebra

5.08 $

Problem: Create a truth table for function \( f=a \& \bar{b} \oplus(b \bigvee c) \). Using the table, write the PDNF and PCNF.

6.3.3 Boolean algebra

2.54 $

Problem: The table shows the numbers of sets of arguments (in lexicographic order), on which the logical function takes a value equal to one. Write this function in PDNF, PCNF, PPNF. Minimize it using the Quine method and the Karnaugh method. \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline \multicolumn{1}{|c|}{ Constituent numbers } \\ \hline 2 & 3 & 6 & 7 & 8 & 14 & 15 & - \\ \hline \end{tabular}

6.3.2 Boolean algebra

5.08 $

Problem: Convert the following formula to CNF: \[ f(p, q, t)=(q \rightarrow(p \rightarrow t)) \rightarrow((q \rightarrow \neg t) \rightarrow(q \rightarrow \neg p)) . \]

6.3.4 Boolean algebra

1.27 $

Problem: Prove the logical identity: \[ \overline{A \vee B}=\bar{A} \wedge \bar{B} \text {. } \]

6.3.5 Boolean algebra

0 $

Problem: Create a truth table, Karnaugh map, perfect disjunctive form (PDNF) and a corresponding combinational circuit for the following options of a logical function (LF), given by a list of unit sets: \[ 1,4,5,7,9,13,14,15 \text {. } \]

6.3.7 Boolean algebra

3.81 $

Problem: Construct the truth table, the Karnaugh map, the perfect disjunctive (PDNF) and a corresponding combinational circuit for the following options of logical functions (LF), given by a list of unit sets: \[ 0,2,5,7,12,13,14,15 \text {. } \]

6.3.8 Boolean algebra

3.81 $

Problem: Represent the Boolean function as a PDNF and draw a circuit, realizing this function. \[ y=\overline{x_{1} \vee x_{2} \vee x_{1} \wedge x_{2}} \vee x_{3} . \]

6.3.6 Boolean algebra

2.54 $

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

6.3.9 Boolean algebra

3.05 $

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

6.3.10 Boolean algebra

4.32 $

Problem: Having created the truth tables of the Boolean functions \( f_{1} \) and \( f_{2} \), check them for equivalence and duality. \[ f_{1}=x \&(y \sim z) ; f_{2}=((x \& y) \sim(x \& z)) \sim x . \]

6.3.11 Boolean algebra

1.52 $

Problem: Using the principle of duality, construct a formula that implements the function dual to the function \( f \), and make sure that the resulting formula is equivalent to the formula \( g \). \[ \begin{array}{l} f=(x \vee y \vee \bar{z}) \& \bar{t} \vee y \vee z ; \\ g=(\bar{x} \vee y \vee z) \overline{\&} t \vee x \& y \overline{\&} z . \end{array} \]

6.3.12 Boolean algebra

2.54 $

Problem: List all dummy variables of function \( f \). 1) \( f\left(x^{3}\right)=(10101010) \); 2) \( f\left(x^{4}\right)=(0101111101011111) \).

6.3.13 Boolean algebra

1.27 $

Problem: Construct the disjunctive normal form of function \( f \). 1) \( f\left(x^{4}\right)=(1011010110110101) \); 2) \( f\left(x^{4}\right)=(0101111101011111) \).

6.3.14 Boolean algebra

2.03 $

Problem: Construct the conjunctive normal form of function \( f \). 1) \( f\left(x^{4}\right)=(1001011100011010) \); 2) \( f\left(x^{4}\right)=(1011000110101110) \).

6.3.15 Boolean algebra

2.03 $

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