Consider the operations $$f(X,Y,Z) = X^{\prime}YZ + XY^{\prime} + Y^{\prime}Z^{\prime} \quad \text{and} \quad g(X,Y,Z) = X^{\prime}YZ + X^{\prime}YZ^{\prime} + XY.$$ Which one of the following is correct?

Let us analyze the given Boolean functions for functional completeness:

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

   $$g(X, Y, Z) = X^{\prime}YZ + X^{\prime}YZ^{\prime} + XY = X^{\prime}Y(Z + Z^{\prime}) + XY = X^{\prime}Y + XY = (X^{\prime} + X)Y = Y$$

   Since $g(X, Y, Z)$ simply reduces to the single variable $Y$ (a projection operation), it cannot generate negation or combinations of multiple variables. Hence, $\{g\}$ is not functionally complete.

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

   $$f(X, Y, Z) = X^{\prime}YZ + XY^{\prime} + Y^{\prime}Z^{\prime} = X^{\prime}YZ + Y^{\prime}(X + Z^{\prime})$$

   • Setting $Y = 0$: $f(X, 0, Z) = 0 + 1 \cdot (X + Z^{\prime}) = X + Z^{\prime}$

   • Setting $X = 0$ in the above: $f(0, 0, Z) = 0 + Z^{\prime} = Z^{\prime}$ (generates the NOT operation)

   • Since the NOT operation is available, we can invert $Z^{\prime}$ to get $Z$, allowing us to form $X + Z$ (the OR operation).

   Because $\{f\}$ can realize both NOT and OR, it forms a functionally complete set.

Thus, only $\{f\}$ is functionally complete [p. 15, Q. 43].