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:- 

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:-