The Gateway to Computer Science Excellence

First time here? Checkout the FAQ!

x

+19 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**?

- Only $P$ and $Q$ are valid.
- Only $Q$ and $R$ are valid.
- Only $P$ and $R$ are valid.
- All $P$, $Q$, $R$ are valid.

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.

+24 votes

Best answer

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')' (Since, AA = A)

= 1 (Since A + A' = 1)

So, R is also valid.

Hence, D choice.

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')' (Since, AA = A)

= 1 (Since A + A' = 1)

So, R is also valid.

Hence, D choice.

- All categories
- General Aptitude 1.2k
- Engineering Mathematics 4.7k
- Digital Logic 1.9k
- Programming & DS 3.5k
- Algorithms 3k
- Theory of Computation 3.7k
- Compiler Design 1.5k
- Databases 2.8k
- CO & Architecture 2.5k
- Computer Networks 2.9k
- Non GATE 837
- Others 1.2k
- Admissions 284
- Exam Queries 397
- Tier 1 Placement Questions 17
- Job Queries 51
- Projects 7

33,705 questions

40,253 answers

114,340 comments

38,861 users