Let, $x_1 \oplus x_2 \oplus x_3 \oplus x_4 = 0$ where $x_1, x_2, x_3, x_4$ are Boolean variables, and $\oplus$ is the XOR operator. Which one of the following must always be TRUE?

Given the equation:

Using the commutative and associative properties of the XOR operation:

Now consider option (C), $\bar{x}_1 \oplus \bar{x}_3 = \bar{x}_2 \oplus \bar{x}_4$: Note that complementation on both inputs to an XOR function leaves the result unchanged:

Therefore:

Since $x_1 \oplus x_3 = x_2 \oplus x_4$, we have:

Thus, (C) is always TRUE.

Counterexample for options (A), (B), and (D): Let $x_1 = x_2 = x_3 = x_4 = 1$. The given condition $1 \oplus 1 \oplus 1 \oplus 1 = 0$ is satisfied, but:

• (A) $x_1 x_2 x_3 x_4 = 1 \neq 0$ (False)

• (B) $x_1 x_3 + x_2 = 1 + 1 = 1 \neq 0$ (False)

• (D) $x_1 + x_2 + x_3 + x_4 = 1 \neq 0$ (False)