Question

4. Using DeMorgan’s Law, write an expression for the complement of F if F(x, y, z) = (x’ + y)(x + z)(y’ + z)’.

The Essentials Of Computer Organization And Architecture (4th Edition) - Chapter 3 - PROB 4E

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Answer 1

DeMorgan's Law says (a + b)'=a'.b', and (a.b)'=a'+b'. + and . are logical AND and OR operators respectively.

Given F= (x’ + y)(x + z)(y’ + z)’.

Applying DeMorgan's law F'=((x’ + y)(x + z)(y’ + z)’)' = (x’ + y)' + ((x + z)(y’ + z)’)'. Note we have applied the second law mentioned above, that is, (a.b)'=a'+b', and I have taken a=(x’ + y) and b=(x + z)(y’ + z)’.

So, F' = (x’ + y)' + ((x + z)(y’ + z)’)'. Now apply the first law on the first term in this expression and apply the second law on the second term in this expression.

So we get, F' = x.y' + (x+z)' + (y'+z) = x.y'+x'.z'+y'+z

y' is common in the first and third terms. So take y' outside.

F'=y'(x+1)+x'.z'+z = y'.1+x'.z'+z= y'+x'.z'+z. We can't simplify the expression further.

Therefore, F' = y'+x'.z'+z is the complement of F.