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# GATE | GATE-CS-2007 | Question 85

• Difficulty Level : Medium
• Last Updated : 27 Sep, 2021

Define the connective * for the Boolean variables X and Y as: X * Y = XY + X’ Y’. Let Z = X * Y.

```Consider the following expressions P, Q and R.
P: X = Y⋆Z
Q: Y = X⋆Z
R: X⋆Y⋆Z=1```

Which of the following is TRUE?
(A) Only P and Q are valid
(B) Only Q and R are valid.
(C) Only P and R are valid.
(D) All P, Q, R are valid.

Explanation: * is nothing but working as EX NOR here.Explanation:

P:

```
X= Y * Z

=(Y XOR Z)’

=YZ + Y’Z’

=Y(XY + X’Y’)+Y’(XY+X’Y’)’

=XY+Y’((Y XOR X)’)’

=XY+Y’(Y XOR X)

=XY+Y’(Y’X+X’Y)

=XY+Y’X

=X(Y+Y’)

=X ```

Q:

```
Y=X*Z

=(X XOR Z)’

=X(XY + X’Y’) + X’(XY + X’Y’)’

=XY+X’(X’Y+XY’)

=XY+X’Y

=Y ```

R:

```
X * Y *Z

WE HAVE SEEN FROM P Y*Z =X

SO X * X

NOT(X XOR X)=X’X’+XX
1
```

SO ALL P,Q,R ARE CORRECT

ANS IS (D)

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