Question 16

Let us examine each option:

• Option A: $(x \oplus y) \oplus z = x \oplus (y \oplus z)$ is the associative property of the XOR operation, which holds universally.
• Option B: Consider $x = 1, y = 1, z = 0$:
  $$\text{LHS} = (1 + 1) \oplus 0 = 1 \oplus 0 = 1$$
  $$\text{RHS} = 1 \oplus (1 + 0) = 1 \oplus 1 = 0$$
  Since $\text{LHS} \ne \text{RHS}$, $(x + y) \oplus z = x \oplus (y + z)$ is NOT a valid identity.
• Option C: $x \oplus y = x'y + xy'$. If $xy = 0$, then $x + y = (x + y)(x + y)' + (x + y) = x'y + xy' = x \oplus y$. This identity is valid.
• Option D: $(xy + x'y')' = (x \odot y)' = x \oplus y$, which is the standard dual relationship between XNOR and XOR.

Thus, Option B is not a valid identity.