GATE CS 2000 Q2.10 — Boolean Algebra

We can solve the simultaneous Boolean equations systematically:

1. Analyze Equation (4): $$xy + \bar{z}\bar{w} = 0$$ For the sum of two Boolean terms to be $0$, both terms must be individually equal to $0$:
   • $xy = 0$ (which is also Equation 2)
   • $\bar{z}\bar{w} = 0 \implies \overline{z + w} = 0 \implies z + w = 1$
   This tells us that at least one of $z$ or $w$ must be $1$.
2. Eliminate Options:
   • Option A ($0\ 1\ 0\ 0$): Here, $z = 0$ and $w = 0$, which gives $z + w = 0$. (Incorrect)
   • Option B ($1\ 1\ 0\ 1$): Here, $x = 1$ and $y = 1$, which gives $xy = 1 \neq 0$. (Incorrect)
   • Option D ($1\ 0\ 0\ 0$): Here, $z = 0$ and $w = 0$, which gives $z + w = 0$. (Incorrect)
3. Verify Option C ($1\ 0\ 1\ 1$): Substituting $x = 1, y = 0, z = 1, w = 1$ into the original equations:
   • $x + y + z = 1 + 0 + 1 = 1$   (True)
   • $xy = 1 \cdot 0 = 0$   (True)
   • $xz + w = (1 \cdot 1) + 1 = 1$   (True)
   • $xy + \bar{z}\bar{w} = 0 + (0 \cdot 0) = 0$   (True)
   All equations are perfectly satisfied.

Thus, the correct solution is C.