Q.State and prove De Morgan's First and Second theorems.

Step 1 (First theorem statement). The complement of the SUM of two logical inputs equals the PRODUCT of their individual complements: $\overline{A+B}=\overline{A}\cdot\overline{B}$.

Step 2 (First theorem proof). Tabulate all four (A,B) combinations and evaluate both sides independently: for (0,0), $\overline{A+B}=\overline{0}=1$ and $\overline{A}\cdot\overline{B}=1\cdot1=1$; for (0,1) and (1,0), both sides evaluate to 0; for (1,1), both sides evaluate to 0. Since both sides match in every row, the identity is proved. Because the left side is the NOR equation and the right side a bubbled-AND equation, this also proves a NOR gate is functionally identical to a bubbled AND gate.

Step 3 (Second theorem statement). The complement of the PRODUCT of two logical inputs equals the SUM of their individual complements: $\overline{A\cdot B}=\overline{A}+\overline{B}$. …

State each theorem, prove it with a full truth-table row-by-row comparison, and n …

• Asserting the theorems without actually working through the truth-table comparison that constitutes the proof. …