Using DeMorgan's Law, write an expression for the complement of F if
F(w, x, y, z) = xyz'(y'z + x)' + (w'yz + x')
F' = (xyz'(y'z + x)' + (w'yz + x'))'
= (xyz'(y'z + x)')' * (w'yz + x')'
= ( (xyz')' + (y'z + x) ) * ( (w'yz)' * x )
= ( ( (xy)' + z ) + (y'z + x) ) * ( ( w + (yz)' ) * x )
= ( ( (x' + y') + z ) + (y'z + x) ) * ( ( w + ( y' + z') ) * x )
= x' + y' + z + (y'z + x)(w + y' + z')x
where this is the appropriate solution given.
But now if we first start like this:
F = xyz'(y'z + x)' + (w'yz + x')
= xyz'((y'z)'x') + (w'yz + x')
= xyz'(((y')' + z')x') + (w'yz + x')
= xyz'((y + z')x') + (w'yz + x')
= xyz'(x'y + x'z') + (w'yz + x')
= xy(x'yz' + x'z') + (w'yz + x')
= x(x'yz' + x'yz') + (w'yz + x')
= xx'yz' + xx'yz' + (w'yz + x')
= xx'yz' + (w'yz + x')
= 0(yz') + (w'yz + x')
F'= (w'yz + x')'
= (w + y' + z')x
and given that
x' + y' + z + ((y'z + x)((w + y' + z')x))
= ((x' + y' + z) + (y'z + x))((x' + y' + z) + ((w + y' + z')x))
= (1 + y' + z + y'z) * ((x' + y' + z) + ((w + y' + z')x))
= (x' + y' + z) + ((w + y' + z')x)
= ((x' + y' + z) + (w + y' + z')) * ((x' + y' + z) + x)
= x' + y' + z + w + y' + z'
= 1
but
(w + y' + z')x != 1
then what have I done that causes the inequality???