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In a test paper, a question is given as :

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)  x1'^x3' = x2'^x4'
(D) x1+x2+x3+x4 = 0

+ is OR and concatenation is AND operator.

Correct answer is C, which I guessed because when all booloean variables are 1, then, neither A,B, or D will hold,But what is the correct way to get to option C?

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(I use the symbol $\oplus$ here because $\wedge$ is mathematically used for conjunction ("and") instead of exclusive-or)

First we rearrange the terms, \begin{align} x_1 \oplus x_2 \oplus x_3 \oplus x_4 &= 0 \\ x_1 \oplus x_3 \oplus x_2 \oplus x_4 &= 0 \\ x_1 \oplus x_3 &= x_2 \oplus x_4 \end{align} Then use $a \oplus b = a' \oplus b'$ to get (C). $$ x_1' \oplus x_3' = x_2' \oplus x_4' $$

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  • $\begingroup$ how to get symbol you mentioned? It is not present on my keyboard. $\endgroup$ – Sumeet May 28 '16 at 14:57
  • $\begingroup$ You have to use LaTeX code which is rendered using MathJax on this site. $\endgroup$ – user51117 May 28 '16 at 15:00
  • $\begingroup$ Whats the procedure for that? $\endgroup$ – Sumeet May 28 '16 at 15:02
  • $\begingroup$ @SumeetSingh you may check out meta.math.stackexchange.com/questions/5020/…. $\endgroup$ – kennytm May 28 '16 at 15:02
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Xor means when we get two same variables answer is 0. That is 0^0=0 and 1^1=0. it is possible to get x1^x2^x3^x4=0 if and only if x1^x2==x3^x4

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