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

De Morgan's two theorems relate the complement operation to AND and OR. The FIRST theorem: the complement of a sum equals the product of the complements, $\overline{A+B}=\overline{A}\cdot\overline{B}$ -- so a NOR gate ($\overline{A+B}$) is functionally identical to a bubbled AND gate. The SECOND theorem: the complement of a product equals the sum of the complements, $\overline{A\cdot B}=\overline{A}+\overline{B}$ -- so a NAND gate ($\overline{A\cdot B}$) is functionally identical to a bubbled OR gate. Both are proved by direct truth-table comparison (evaluating both sides for all four input combinations and checking the output columns match row for row). Their practical consequence is that NAND alone, or NOR alone, is 'universal' -- since either can reproduce a NOT gate (tie both inputs together) and, via the theorems, the other two basic gates as well, any digital logic function can be built from NAND gates only, or from NOR gates only. …