Consider the operations \(\textit{f (X, Y, Z) = X'YZ + XY' + Y'Z'}\) and…

Key steps and conclusion:

Simplify the second function:

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

So the second function is just the projection Y. A single-variable projection cannot be functionally complete.

Now analyze the first function for functional completeness using Post's criteria (a single function is functionally complete if it is not contained in any of the five closed classes: preserves 0, preserves 1, monotone, affine, self-dual).

• Does not preserve 0: f(0,0,0) = X'YZ + XY' + Y'Z' = 0 + 0 + 1 = 1, so f(0,0,0) = 1 ≠ 0.

• Does not preserve 1: f(1,1,1) = 0 + 0 + 0 = 0, so f(1,1,1) = 0 ≠ 1.

• Not monotone: compare inputs (0,0,0) and (0,1,0). We have f(0,0,0)=1 but f(0,1,0)=0, so increasing an input caused the output to decrease.

• Not self-dual: take input (0,0,1) where f(0,0,1)=0, and its complement (1,1,0) where f(1,1,0)=0. Self-duality would require f(0,0,1)=¬f(1,1,0)=1, so self-duality fails.

• Not affine: the expression contains product (AND) terms such as X'Y Z, so its algebraic normal form has degree >1; affine functions have degree ≤1.

Because the first function fails each of the five preservation/closure properties, it is functionally complete. The second function reduces to the projection Y and is not functionally complete. Therefore, only the first function is functionally complete.