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 is TRUE?

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 is TRUE? Correct Answer x̅₁ ⊕ x̅₃ = x̅₂ ⊕ x̅₄

Concept:

XOR gate is a gate that gives a true output when the number of true inputs is odd.

Explanation:

Given, x₁ ⊕ x₂ ⊕ x₃ ⊕ x₄ = 0

Where, x₁, x₂, x₃, x₄ are Boolean variables, and ⊕ is the XOR operator

Consider x₁ = 1, x₂ =1, x₃ =1 and x₄= 1

1 ⊕ 1 ⊕ 1 ⊕ 1 = 0

Now, consider all the options one by one.

1) x₁x₂x₃x₄ = 0

Here, put the value of x₁,x₂, x₃, x₄ as 1

So, 1.1.1.1 = 1

2) x₁x₃ + x₂ = 0

1.1 + 1 =1

3) x̅₁ ⊕ x̅₃ = x̅₂ ⊕ x̅₄

Here, x̅₁ = x̅₃ = x̅₂ = x̅₄ = 0,

So, 0 ⊕ 0 = 0 ⊕ 0,

0 = 0

4) x₁ + x₂ + x₃ + x₄ = 0

As, 1+1+1+1 = 1