8,761 views
42 votes
42 votes

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?

  1. Only $P$ and $Q$ are valid.
  2. Only $Q$ and $R$ are valid.
  3. Only $P$ and $R$ are valid.
  4. All $P$, $Q$, $R$ are valid.

5 Answers

Best answer
53 votes
53 votes
P:

$Y * Z = Y * (X*Y)$
$= Y * (XY + X'Y')$
$= Y (XY + X'Y') + Y' (XY + X'Y')'$
$= XY + Y' ((X' + Y') (X+ Y))$
$= XY + Y' (X'Y + XY')$
$= XY + XY'$
$= X(Y + Y')$
$= X$

So, P is valid.

Q:

$X * Z = X * (X*Y)$
$= X * (XY + X'Y')$
$= X (XY + X'Y') + X' (XY + X'Y')'$
$= XY + X' ((X' + Y') (X+ Y))$
$= XY + X' (X'Y + XY')$
$= XY + X'Y$
$= Y(X + X')$
$= Y$

So, Q is also valid.

R:

$X * Y * Z = (X * Y) * (X * Y)$
$= (XY + X'Y') * (XY + X'Y')$
$= (XY + X'Y') (XY + X'Y') + (XY + X'Y')' (XY + X'Y')'$
$= (XY + X'Y') + (XY + X'Y')' (\because AA = A)$
$= 1 (\because A + A' = 1)$

So, R is also valid.

Hence, D choice.
edited by
22 votes
22 votes

P: rhs=Y*Z =Y*(X*Y)=(Y*Y)*X=1*X=X=lhs  so valid

(we can use the fact that X*Y IS EQUIV TO X XNOR Y which is associative AND commutative)

Q:rhs=X*Z =X*(X*Y)=(X*X)*Y=1*Y=Y=lhs  so valid

R:X*Y*Z=X*Y*(X*Y) =(X*X)*(Y*Y)=1 so valid

3 votes
3 votes

 

First of all here * operation represents XNOR.

P: X=Y∗Z = Y XNOR Z = Y xnor X xnor Y = Y xnor Y xnor X = 1 xnor X = X

Q:Y=X∗Z = X xnor X xnor Y = Y

R:X∗Y∗Z=1 

   LHS = X xnor Y xnor X xnor Y = 1 xnor 1 = 1

All are valid.

0 votes
0 votes

The given operation * is XNOR. The law of XNOR : If A XNOR B = C then A XNOR C = B. This can also be verified by solving the equation. Just solve P and that will lead to X=X. So if P holds True, using the law Q also holds True(swap(Y,X)) gives the same output. Therefore P and Q holds.

Now coming to R:. XNOR has the property of odd number of 1’s detector(if inputs are odd) and even number of 1’s detector(if input is even). R: X(exnor)Y(exnor)Z = 1 it holds True, since the number of inputs are 3. 

Therefore D is the answer

Answer:

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