6.3.1 Boolean algebra

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. \[ \begin{array}{l} ((a \wedge \bar{c}) \downarrow(b \wedge \bar{c})) \wedge((a \mid d)(\overrightarrow{b \wedge d}))= \\ =((a \mid b) \mid(a \oplus \bar{b})) \rightarrow((c \oplus d) \wedge(d \rightarrow c)), \text { option } 2 . \end{array} \]