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Let x1 ⊕ x2 ⊕ x3 ⊕ x4 = 0 where x1, x2, x3, x4 are Boolean variables, and ⊕ is the XOR operator. Which one of the following must always be TRUE?

1. x1 x2 x3 x4 = 0
2. x1 x3 + x2 = 0
3. x̄1 ⊕ x̄3 = x̄2 ⊕ x̄4
4. x1 + x2 + x3 + x4 = 0

Correct answer: x̄1 ⊕ x̄3 = x̄2 ⊕ x̄4

Solution

The equation x1 ⊕ x2 ⊕ x3 ⊕ x4 = 0 indicates that the number of true values among x1, x2, x3, and x4 must be even. The correct option shows that the negations of the variables also maintain this even parity, as the XOR operation on the negated variables reflects the same relationship.

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