Complement of a Boolean Function

The function is F and it’s complement is F’. Suppose there is a function as follows

F = x' y  z' + x' y' z

We can find the complement of the function using two rule stated by DeMorgan’s Law.

1. Change the OR gates with AND gates or change the AND gates with OR gates.
2. Change each literal of the function with it’s complement.

Now, we will apply the above two principle in F to obtain F’.

F'= (x' y z' + x' y 'z)'

       = (x + y' + z).(x + y + z')

Consider more examples,

F = x y' + x' y

F' = (x y' + x' y)

   = (x' + y).(x + y)

Source: Digital Logic and Computer Design by Morris Mano

F  = (AB' + C)D' + E

F' = ((AB' + C)D' + E)'

   = ((AB' + C)' + D)(E')

   = ((A' + B ) . C') + D).(E')

   = (A' + B + D)(C' + D).(E')

Source: Digital Logic and Computer Design by Morris Mano

F  = (x + y' + z)(x' + z')(x + y)

F' = ((x + y' + z)(x' + z')(x + y))'

   = ((x + y' + z)' + (x' + z')' + (x + y)')'

   =((x' y z')+(x z) + (x' y'))

   = x' y z' + x z + x' y'

Source: Digital Logic and Computer Design by Morris Mano