DeMorgan's Law 1: ( X̅ . Y̅ ) is compliment of ( X + Y )

We have to prove –

(i)                ( X + Y ) + ( X̅ . Y̅ ) = 1 and

(ii)              ( X + Y ) . ( X̅ . Y̅ ) = 0

 

 ( X + Y) + ( X̅ . Y̅) = 1

( X + Y ) + ( X̅ . Y̅ ) = [ ( X + Y ) + X̅ ] . [ ( X + Y ) + Y̅ ]

                           = ( 1 + Y )  . ( 1 + X )

                           = 1 . 1

                           = 1

( X + Y ) . ( X̅ . Y̅ ) = 0

( X + Y ) . ( X̅ . Y̅ ) = X̅ . Y̅ . X + X̅ . Y̅ . Y

                             = 0 + 0

                             = 0

( X̅ . Y̅ ) is compliment of ( X + Y ).

So we can write it as :-