Question 49

1. Analyze $g(X, Y, Z)$:

   $$g(X, Y, Z) = X' Y Z + X' Y Z' + X Y$$

   Factor out $X' Y$:

   $$g(X, Y, Z) = X' Y (Z + Z') + X Y = X' Y(1) + X Y = (X' + X) Y = Y.$$

   The function $g(X, Y, Z)$ simplifies identically to the single variable $Y$. It is degenerate and cannot invert or combine signals. Hence, $\{g\}$ is not functionally complete.

2. Analyze $f(X, Y, Z)$:

   $$f(X, Y, Z) = X' Y Z + X Y' + Y' Z' = X' Y Z + Y'(X + Z').$$
   • To synthesize NOT: set $X = Z$:
     $$f(X, Y, X) = X' Y X + Y'(X + X') = 0 + Y'(1) = Y'.$$
     Thus, $f$ can generate the NOT gate $\text{NOT}(Y) = f(X, Y, X)$.
   • To synthesize AND/OR: set $Y = 1$:
     $$f(X, 1, Z) = X'(1)Z + (0) + (0) = X' Z.$$
     Combining $X' Z$ with the NOT gate generates $X Z$ (the AND gate).
     Since $\{f\}$ can generate both NOT and AND, $\{f\}$ is functionally complete.

Therefore, only $\{f\}$ is functionally complete (option B).