# GATE | GATE-CS-2000 | Question 33

• Last Updated : 19 Nov, 2018

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

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

Explanation: We solve this question by putting in the options in the statements.

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Statement 1 :  x + y + z = 1

```OPTION    x  y  z  w    LHS     LHS=1
--------------------------------------
A       0  1  0  0     1       Yes
B       1  1  0  1     1       Yes
C       1  0  1  1     1       Yes
D       1  0  0  0     1       Yes
```

Till now, all the options are possible.

Statement 2 : x y = 0

```OPTION    x  y  z  w    LHS     LHS=0
--------------------------------------
A       0  1  0  0     0       Yes
B       1  1  0  1     1       No
C       1  0  1  1     0       Yes
D       1  0  0  0     0       Yes
```

Since LHS ≠ 0, B is not possible.

Statement 3 : x z + w = 1

```OPTION    x  y  z  w    LHS     LHS=1
--------------------------------------
A       0  1  0  0     0       No
C       1  0  1  1     1       Yes
D       1  0  0  0     0       No
```

Since LHS ≠ 1, A and D are not possible.

Statement 4 : x y + z’ w’ = 0

```OPTION    x  y  z  w    LHS    LHS=0
--------------------------------------
C       1  0  1  1     0      Yes

```

Thus, C is the correct option.
Please comment below if you find anything wrong in the above post.

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