GATE CSE 1997 | Question: 2-1

Question

Let $*$ be defined as $x * y = \bar{x} + y$. Let $z = x * y$. Value of $z * x$ is 

• A. $\bar{x} + y$
• B. $x$
• C. $0$
• D. $1$

Comments

opton a)

$z*x=z+x=(x*y)+x=x+y+x=x+y$

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z + x= x+y+x = x+y

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Answer 1

Answer is option B.

$z* x = {(x*y)} * x$

$=\left(\bar{x} + y\right) * x$

$=\overline{\bar{x} + y} + x $

$x.\bar{y} + x = x$

Comments

my pleasure sir

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@Arjun

Sir, since nature of * is not given but if we assume nature of * is right-associative then,

$z*x = x * y *x.$

$  z = x * (y * x)$

$ z = x * (\bar{ y} + x)$

$z = \bar{x} + \bar{y} + x$

$z = 1 + \bar{y} = 1.$

Please tell me where am I doing mistake?

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Sir, since nature of * is not given but if we assume nature of * is right-associative then,………

Please tell me where am I doing mistake?

The given operation $*$ is Not associative, So that’s the mistake.

Note that $*$ is Implication operation, i.e. $x*y = x \rightarrow y$, and Implication operation is NOT associative.

Now, we can write $z*x = (z)*x = (x*y)*x.$

Also note that Implication operation is NOT Commutative.

Answer 2

The answer is Option (b).
Given
$x*y = x' +y$

$z= x*y$

Therefore $z = x'+y$

$z*x = z' +x$
       $= (x'+y)' +x$
       $= x.y' + x$
       $= x(y' +1)$
       $= x$
Answer $z*x = x$

Comments

He asked something different. He didn't put that compliment in this question .

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Question Edited, now the question is correct.

Answer 3

Z*X=(X*Y)*X

=(X^c+Y)^c+X

=X.Y^c+X

=X(1+Y^c)

=X

Comments

Which rule is applied to change from step 1 to 2 ?

Answer 4

Answer : Option B