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State and Prove DeMorgan's Laws.
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DeMorgan's Theorems:-

• A mathematician named DeMorgan developed a pair of important rules regarding group complementation in Boolean algebra.

Theorem 1:-

• DeMorgan's First theorem states that for any two elemnts A and B in a boolean algebra, the complemeny of a product is equal to the sum of complements.

$\overline {AB} = \overline A +\overline B$

• The LHS of this theorem represents a NAND gate with inputs A and B, whereas the RHS represents an OR gate with inverted inputs.
• This OR gate is called as Bubbled OR.
• Thus according to DeMorgan's First theorem NAND gate is equaivalent to Bubbled OR gate.
• The circuit representation is given below

Proof:-

A B $\overline{AB}$ $\overline A$ $\overline B$ $\overline A + \overline B$
0 0 1 1 1 1
0 1 1 1 0 1
1 0 1 0 1 1
1 1 0 0 0 0

Theorem 2:-

• DeMorgan's Second theorem states that for any two elemnts A and B in a boolean algebra, the complemeny of a sum is equal to the product of complements.

$\overline {A+B} = \overline A \ \overline B$

• The LHS of this theorem represents a NOR gate with inputs A and B, whereas the RHS represents an AND gate with inverted inputs.
• This AND gate is called as Bubbled AND.
• Thus according to DeMorgan's Second theorem NOR gate is equaivalent to Bubbled AND gate.
• The circuit representation is given below

Proof:-

A B $\overline{A+B}$ $\overline A$ $\overline B$ ${\overline A \ \ \overline B}$
0 0 1 1 1 1
0 1 0 1 0 0
1 0 0 0 1 0
1 1 0 0 0 0