STATEMENT FOR DE MORGAN'S FIRST THEOREM*:*The complement of a logical sum equals the logical product of the                                                                                                       complements.  

LOGICAL EQUATION*:*  (A+B)' = A'+B'

PROOF :

CASE 1 :Let A=0 & B=0

LHS = (A+B)' = (0+0)' = 0' =1

RHS = A'.B' = 0',0' = 1.1 =1

CASE 2 :

Let A=0 & B=1

LHS = (0+1)' = 1' = 0

RHS = 0'.1' = 1.0 = 0

CASE 3 :

Let A=1 & B=0

LHS = (1+0)' = 1' = 0

RHS = 1'.0' = 0.1 = 1 

CASE 4 :

Let A =1 & B=1 

LHS = (1+1)' = 1' = 0 

RHS = 1'.1' = 0.0 = 0

As in all 4 possible cases , LHS = RHS , First theorem is proved.

                 STATEMENT FOR DE MORGAN'S SECOND THEOREM :

The complement of a logic product equals the logical sum of the complements .

LOGIC EQUATION : (A.B)' = A'+B'

PROOF :

CASE 1 : Let A = 0 , & B= 0 

LHS = (A.B)' = (0.0)' = 0'

         = 1

RHS = A'+B' = 0'+0' = 1+1 = 1

CASE 2 :

Let A =0 & B=1

LHS = (0.1)'=0' = 1

RHS = 0'+1' = 1+0 = 1

CASE 3 :Let A = 1 & B= 0

LHS = (1.0)' = 0' = 1

RHS = 1'+0' = 0+1 = 1

CASE 4 : Let A=1 & B=1

LHS = (1.1)' = 1' = 0

RHS = 1'+1' = 0+0 = 0

As in all 4 possible cases , LHS=RHS , DE MORGAN'S THEORM was proved .