GATE CS 2016 Set 2 — Question 18
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Digital Logic → Boolean Algebra & Logic Gates → Boolean Laws & Theorems
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Question
Let, where are Boolean variables, and is the XOR operator.
Which one of the following must always be TRUE?
Which one of the following must always be TRUE?
Correct answer
(C) x₁ ⊕ x₃ = x₂ ⊕ x₄
Solution
Given the equation:This implies that the number of variables with value 1 is even (0, 2, or 4).Let's analyze the options:(A)
This expression is false if all variables are 1 (). Since , the condition holds, but the option statement is false. Thus, (A) is not always true.(B)
Consider the case where . The XOR sum is , so the condition holds. However, . Thus, (B) is not always true.(C)
Recall that .
So, the LHS becomes and the RHS becomes .
The statement is equivalent to .
From the given equation , we can rearrange terms:This matches the simplified form of option (C). Thus, (C) is always true.(D)
This implies all variables are 0. However, the condition also holds if all variables are 1 (sum = 4) or if two are 1 (sum = 2). Thus, (D) is not always true.
This expression is false if all variables are 1 (). Since , the condition holds, but the option statement is false. Thus, (A) is not always true.(B)
Consider the case where . The XOR sum is , so the condition holds. However, . Thus, (B) is not always true.(C)
Recall that .
So, the LHS becomes and the RHS becomes .
The statement is equivalent to .
From the given equation , we can rearrange terms:This matches the simplified form of option (C). Thus, (C) is always true.(D)
This implies all variables are 0. However, the condition also holds if all variables are 1 (sum = 4) or if two are 1 (sum = 2). Thus, (D) is not always true.
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