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2、已知公式 A 的部分真值表如下:
\begin{tabular}{ccccc}
$p$ & $q$ & $r$ & $s$ & $A$ (其他赋值为 0 ) \
0 & 0 & 0 & 0 & 1 \
0 & 0 & 1 & 0 & 1 \
0 & 1 & 1 & 1 & 1 \
1 & 0 & 0 & 1 & 1 \
1 & 0 & 1 & 0 & 1 \
1 & 1 & 1 & 0 & 1
\end{tabular}
A) 写出其所有极大项与极小项
B) 写出 A 的主析取范式与主合取范式

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∻Solution by Steps∻ ‖ step 1 ⋮ Identify the minterms and maxterms from the given partial truth table. ‖ ‖ step 2 ⋮ Minterms are the rows where $A = 1$. From the table, these are: $(0,0,0,0)$, $(0,0,1,0)$, $(0,1,1,1)$, $(1,0,0,1)$, $(1,0,1,0)$, $(1,1,1,0)$. ‖ ‖ step 3 ⋮ Maxterms are the rows where $A = 0$. Since all given rows have $A = 1$, there are no maxterms in the provided table. ‖ ‖ step 4 ⋮ Write the minterms in standard form: $m_0 = \overline{p} \overline{q} \overline{r} \overline{s}$, $m_2 = \overline{p} \overline{q} r \overline{s}$, $m_7 = \overline{p} q r s$, $m_9 = p \overline{q} \overline{r} s$, $m_{10} = p \overline{q} r \overline{s}$, $m_{14} = p q r \overline{s}$. ‖ ‖ step 5 ⋮ The Principal Disjunctive Normal Form (PDNF) is the sum of minterms: $A = m_0 + m_2 + m_7 + m_9 + m_{10} + m_{14}$. ‖ ‖ step 6 ⋮ The Principal Conjunctive Normal Form (PCNF) is the product of maxterms. Since there are no maxterms given, we consider the complement of the minterms. ‖ ‖ step 7 ⋮ Write the complement of each minterm: $M_0 = p + q + r + s$, $M_2 = p + q + \overline{r} + s$, $M_7 = p + \overline{q} + \overline{r} + \overline{s}$, $M_9 = \overline{p} + q + r + \overline{s}$, $M_{10} = \overline{p} + q + \overline{r} + s$, $M_{14} = \overline{p} + \overline{q} + \overline{r} + s$. ‖ ‖ step 8 ⋮ The PCNF is the product of these maxterms: $A = M_0 \cdot M_2 \cdot M_7 \cdot M_9 \cdot M_{10} \cdot M_{14}$. ‖ ∻Answer∻ ⚹ The minterms are $m_0, m_2, m_7, m_9, m_{10}, m_{14}$. The maxterms are $M_0, M_2, M_7, M_9, M_{10}, M_{14}$. The PDNF is $A = m_0 + m_2 + m_7 + m_9 + m_{10} + m_{14}$. The PCNF is $A = M_0 \cdot M_2 \cdot M_7 \cdot M_9 \cdot M_{10} \cdot M_{14}$. ⚹ ∻Key Concept∻ ⚹Minterms and Maxterms⚹ ∻Explanation∻ ⚹Minterms are the rows where the function is true (1), and maxterms are the rows where the function is false (0). The PDNF is the sum of minterms, and the PCNF is the product of maxterms.⚹◊What is the process for finding the minterms and maxterms of a Boolean function given a partial truth table?⍭ Generate me a similar question◊

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