DeMorgan's Law 2 : ( 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 + ( X̅ + Y̅) ] . [ Y  + ( X̅ + Y̅ ) ]

                           = ( 1 + Y̅ )  . ( 1 + X̅ )

                           = 1 . 1

                           = 1

( X . Y ) . ( X̅ + Y̅ ) = 0

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

                             = 0 + 0

                             = 0

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

So we can write it as :-