Discussion
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?
Answer: C
The statement that always holds is \(\bar{x}_{1} \oplus \bar{x}_{3} = \bar{x}_{2} \oplus \bar{x}_{4}\).
XOR-ing \(x_{2} \oplus x_{4}\) onto both sides of \(x_{1} \oplus x_{2} \oplus x_{3} \oplus x_{4} = 0\) gives \(x_{1} \oplus x_{3} = x_{2} \oplus x_{4}\). Complementing both inputs of an XOR does not change it, \(\bar{a} \oplus \bar{b} = a \oplus b\), so
\[ \begin{aligned} \bar{x}_{1} \oplus \bar{x}_{3} &= x_{1} \oplus x_{3} \\ &= x_{2} \oplus x_{4} \\ &= \bar{x}_{2} \oplus \bar{x}_{4} \end{aligned} \]
The others fail for the assignment \(x_{1} = x_{2} = x_{3} = x_{4} = 1\), which satisfies the condition (four ones XOR to 0): then \(x_{1}x_{2}x_{3}x_{4} = 1\), \(x_{1}x_{3} + x_{2} = 1\) and \(x_{1} + x_{2} + x_{3} + x_{4} = 1\) as an OR.
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