GATE CSE 2000 | Question: 2.10

Question

The simultaneous equations on the Boolean variables $x, y, z$ and $w$,

• $x + y + z = 1 $
• $xy = 0$
• $xz + w = 1$
• $xy + \bar{z}\bar{w} = 0$

have the following solution for $x, y, z$ and $w,$ respectively:

• A. $0 \ 1 \ 0 \ 0$
• B. $1 \ 1 \ 0 \ 1$
• C. $1 \ 0 \ 1 \ 1$
• D. $1 \ 0 \ 0 \ 0$

Comments

when i seen first time , its equation like plus(arithemtic )after sometime i undersrtand 1+1=0 but equation give 1 how, then i realize + the plus is boolean OR

---

Correct answer is option C ✅

---

Opction c
just put the options that's it

Answer 1

Take each option one by one and try to put the values of $x, y, z$ and $w$ in question:
1.

• $0+1+0 =1$
• $0.1 =0$
• $0.0+0 =0$  (went wrong)      So, this is not the right option

2.

• $1+1+0 =1$
• $1.1 =1$ (went wrong)          not right

3 .

• $1+0+1 =1$
• $1.0 =0$
• $1.1+1=1$
• $1.0 +1.0 =0 $     This is the right option

Correct Answer: $C$

Comments

Look at each equation and see which option is not satisfying that equation. That option will get eliminated.
Equation 1 is satisfied by all options.
Equation 2 eliminate option B
Equation 3 eliminate options A and D.

---

I think eliminating options is the best way to solve this problem in real exam

Answer 2

My approach option elimination:-
as xy=0
then x= or y= 0 or x and y both are 0. So B is eliminated.

xy+z'w'=0
implies these xy=0 and z'w' =0
implies these either z'=0 or w'=0 or both
impiles these either z=1 or w=1 or both.
So A and D are eliminated.

Answer is C

Answer 3

Answer: C

Comments

x + y + z =  1 + 0 + 1 = 0 ? only right

---

Got it. basic OR gate properties

Answer 4

Answer -    C. 1 0 1 1   * think without ink*

Answer 5

option c is right.

Answer 6

i think we can solve by elimination method very easy .....

   like from 2nd given equation      ( xy =0 ) , we can say ether x or y zero but in option b they given 1101 so B option is wrong

   next equation 4 is    xy+z`w`= 0 so from above we know xy is 0 so now we have only z`w` e.g  (0+z`w`) =z`w`,

  so now we have z`w`=0    now here apply de morgen law .so we get z+w =1.

now we have z+w =1 so z or w must 1 but in option A and D they given both 0 . so we can elminate these A and D option .

so correct option is C