Which one of the following is NOT a valid identity?

Let us examine each option:

• Option A: XOR is associative, so $(x \oplus y) \oplus z = x \oplus (y \oplus z)$ is a valid identity.

• Option B: $(x + y) \oplus z = x \oplus (y + z)$ is not valid. For example, let $x = 1$, $y = 1$, $z = 0$:

  $$\text{LHS} = (1 + 1) \oplus 0 = 1 \oplus 0 = 1$$

  $$\text{RHS} = 1 \oplus (1 + 0) = 1 \oplus 1 = 0$$

  Since $\text{LHS} \neq \text{RHS}$, this is not a valid identity.

• Option C: $x \oplus y = x^{\prime}y + xy^{\prime}$. If $xy = 0$, then $x + y = x^{\prime}y + xy^{\prime} + xy = x^{\prime}y + xy^{\prime} = x \oplus y$. Thus, this is valid.

• Option D: $xy + x^{\prime}y^{\prime}$ is the standard XNOR expression. The complement of XNOR is XOR, so $(xy + x^{\prime}y^{\prime})' = x \oplus y$, which is valid.

Therefore, B is NOT a valid identity.