GATE CSE 2007 | Question: 33

Question

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.

Comments

Here, XNOR gate is given and XNOR gate does not obey associative law than how can someone put  X*Y instead of Z in 'R' ?

X*Y*Z = 0 always (because X*Y=Z means sum of 1's in X and Y is even and if I add Z in it than that will be even no. of 1's and that will make it odd term and is 0 always.)

If it is X*(Y*Z) than it is =1

otherwise for XNOR: x*Y*Z = 0

 

So only P & Q are valid.

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Remarks : XNOR gate is commutative and associative.

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Why can't we simply use the XNOR's property of associativity and commutativity while answering such question?

 

P:Y * Z

= Y * (X*Y)

= Y *(Y*X)       (commutative)

= (Y * Y) * X      (associative)

= 1 * X

= X = LHS

So, P is valid and so on.

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for which input of x , y ,z these conditions in truth table is valid ? 

Z=X∗Y

 

R:X∗Y∗Z=1

for any 1 and 0 that is satisfying  ?

x	y	z	Q
0	0	0	1
0	0	1	0
0	1	0	0
0	1	1	1
1	0	0	0
1	0	1	1
1	1	0	1
1	1	1	0

Answer 1

ans D