###### Nielit Scientist-B IT 22-07-2017

October 14, 2023###### Software-Engineering

October 14, 2023# Boolean-Algebra

Question 6 |

Let, x_{1}⊕x_{2}⊕x_{3}⊕x_{4} = 0 where x_{1}, x_{2}, x_{3}, x_{4} are Boolean variables, and ⊕ is the XOR operator. Which one of the following must always be **TRUE**?

x _{1}x_{2}x_{3}x_{4} = 0 | |

x _{1}x_{3}+x_{2} = 0 | |

x _{1} + x_{2} + x_{3} + x_{4} = 0 |

Question 6 Explanation:

Given expression is,

x

A) x

Put x

The given equation will be zero, i.e.,

1 ⊕ 1 ⊕ 1 ⊕ 1 = 0

But,

x

So, false.

B) x

Put x

The given equation will be zero, i.e.,

1 ⊕ 1 ⊕ 0 ⊕ 0 = 0

But,

x

So, false.

D) x

Let x

The given equation will be zero, i.e.,

1 ⊕ 1 ⊕ 0 ⊕ 0 = 0

But,

x

So, false.

(i) True.

x

_{1}⊕ x_{2}⊕ x_{3}⊕ x_{4}= 0 —–(1)A) x

_{1}x_{2}x_{3}x_{4}= 0Put x

_{1}= 1, x_{2}= 1, x_{3}= 1, x_{4}= 1The given equation will be zero, i.e.,

1 ⊕ 1 ⊕ 1 ⊕ 1 = 0

But,

x

_{1}x_{2}x_{3}x_{4}≠ 0So, false.

B) x

_{1}x_{3}+ x_{2}= 0Put x

_{1}= 1, x_{2}= 1, x_{3}= 0 , x_{4}= 0The given equation will be zero, i.e.,

1 ⊕ 1 ⊕ 0 ⊕ 0 = 0

But,

x

_{1}x_{3}+ x_{2}≠ 0So, false.

D) x

_{1}+ x_{2}+ x_{3}+ x_{4}= 0Let x

_{1}=1, x_{2}=1, x_{3}=0, x_{4}=0The given equation will be zero, i.e.,

1 ⊕ 1 ⊕ 0 ⊕ 0 = 0

But,

x

_{1}+ x_{2}+ x_{3}+ x_{4}≠ 0So, false.

(i) True.

Correct Answer: C

Question 6 Explanation:

Given expression is,

x

A) x

Put x

The given equation will be zero, i.e.,

1 ⊕ 1 ⊕ 1 ⊕ 1 = 0

But,

x

So, false.

B) x

Put x

The given equation will be zero, i.e.,

1 ⊕ 1 ⊕ 0 ⊕ 0 = 0

But,

x

So, false.

D) x

Let x

The given equation will be zero, i.e.,

1 ⊕ 1 ⊕ 0 ⊕ 0 = 0

But,

x

So, false.

(i) True.

x

_{1}⊕ x_{2}⊕ x_{3}⊕ x_{4}= 0 —–(1)A) x

_{1}x_{2}x_{3}x_{4}= 0Put x

_{1}= 1, x_{2}= 1, x_{3}= 1, x_{4}= 1The given equation will be zero, i.e.,

1 ⊕ 1 ⊕ 1 ⊕ 1 = 0

But,

x

_{1}x_{2}x_{3}x_{4}≠ 0So, false.

B) x

_{1}x_{3}+ x_{2}= 0Put x

_{1}= 1, x_{2}= 1, x_{3}= 0 , x_{4}= 0The given equation will be zero, i.e.,

1 ⊕ 1 ⊕ 0 ⊕ 0 = 0

But,

x

_{1}x_{3}+ x_{2}≠ 0So, false.

D) x

_{1}+ x_{2}+ x_{3}+ x_{4}= 0Let x

_{1}=1, x_{2}=1, x_{3}=0, x_{4}=0The given equation will be zero, i.e.,

1 ⊕ 1 ⊕ 0 ⊕ 0 = 0

But,

x

_{1}+ x_{2}+ x_{3}+ x_{4}≠ 0So, false.

(i) True.

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