Question 18

We are given:

Since XOR is commutative and associative, and $a \oplus b = 0 \iff a = b$, we can group the variables:

Recall that the complement of a Boolean variable can be written as $\overline{x} = x \oplus 1$. Thus:

Similarly:

Since $x_1 \oplus x_3 = x_2 \oplus x_4$, it follows immediately that:

This equality holds under all assignments satisfying the premise.

Evaluate the other options with a counterexample:
Let $x_1 = x_2 = x_3 = x_4 = 1$. Then $1 \oplus 1 \oplus 1 \oplus 1 = 0$, but:

• (A) $x_1 x_2 x_3 x_4 = 1 \ne 0$
• (B) $x_1 x_3 + x_2 = 1 + 1 = 1 \ne 0$
• (D) $x_1 + x_2 + x_3 + x_4 = 1 \ne 0$
  All three fail.

Therefore, option (C) must always be true.