State and prove De Morgan's Frist and second theorems.

Explanation

De morgan's First theorem :
The first theorem states that the complement of the sum of two logical inputs is equal to the product of its complements .
Proof :
The Boolean equation for NOR gate is `Y=bar(A+B)`
The Boolean equation for a bubbled AND gate is `Y=bar(A) .bar(B)` Bothe cases generate same outputs for same inputs. It can be verified using the following truth table.

From the above truth , we can conclude `bar(A+B)=bar(A).bar(B).`
It also says that a NOR gate is equal to a bubbled AND gate.
The corresponding logic circuit diagram

De morgan's second theorem :
The second theorem State that the complement of the product of two inputs is equal to the sum of its complements.
Proof :
The Boolean equation for NAND gate is `Y=bar(AB)` Tehe Boolean equation for a bubbled OR gate is `Y=bar(A)+bar(B).`
A and B are the inputs and Y is the output . The above two equations produces the same output for the same inputs.

From the above truth table, we can conclude .
`bar(A+B) = bar (A).bar(B)`
Thus, De Morgan's First Theorem is proved . It also says, a NAND gate is equal to a bubbled OR gate.
The corresponding logic circuit diagram