GATE | GATE-CS-2016 (Set 2) | Question 18

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?

(A) x1x2x3x4 = 0
(B) x1x3+x2 = 0
(C) x′1⊕x′3=x′2⊕x′4
(D) x1+x2+x3+x4 = 0


Answer: (C)

Explanation: First we rearrange the terms,
x1⊕x2⊕x3⊕x4 = 0
x1⊕x3⊕x2⊕x4 = 0
x1⊕x3 = x2⊕x4

Then use a⊕b=a′⊕b′a⊕b=a′⊕b′ to get (C).

x′1⊕x′3=x′2⊕x′4

Another approach :
You can take a counter example to disprove other options.
You can take x1 = x2 = x3 = x4 = 1.
Only option (C) is correct.

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