GATE CSE 2010 | Question: 9

Question

The Boolean expression of the output $f$ of the multiplexer shown below is

• A. $\overline {P \oplus Q \oplus R}$
• B. $P \oplus Q \oplus R$
• C. $P+Q+R$
• D. $\overline{P+Q+R}$

Comments

For those who are confused between A and B.

For odd number of inputs (P xor Q xor R) = (P xnor Q xnor R).

But it is never possible to have (P xor Q xor R) = (P xor Q xor R)' , Doesn't matter number of inputs are odd or even, they will never be equal.

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Detailed Video Solution: https://youtu.be/wEhjAAs8iqk?t=480&feature=shared 

Answer 1

$f = S_0'S_1' R + S_0'S_1R' + S_0S_1'R' + S_0S_1R$
$\quad=Q'P'R + Q'PR' + QP'R' + QPR $
$\quad= Q'(P⊕R) + Q(P⊕R)' $
$\quad= Q⊕P⊕R = P⊕Q⊕R$

Doing truth value substitution,
$${\begin{array}{|cccc|c|}\hline
\textbf{P}&    \textbf{Q}&  \textbf{R}&\bf{f}& \bf{P \oplus Q \oplus R } \\\hline
0&0&0&0&0 \\ 0&0&1&1&1\\     0&1&0&1&1\\    0&1&1&0&0\\    1&0&0&1&1\\  1&0&1&0& 0  \\    1&1&0&0&0\\    1&1&1&1& 1\\ \hline
\end{array}}$$ Correct Answer: B

Comments

 

Q′Q'(P⊕R)+Q(P⊕R')′

HOW TO SIMPLIFY IT:

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@Arjun sir, dosent the answer hold good for option A too?? as we have odd number of variables , xor and xnor give same ans right ?

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Let : Y = P XOR R

So : Q' Y + Q Y' 

= Q XOR Y

= Q XOR P XOR R

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@Arjun sir, dosent the answer hold good for option A too?? as we have odd number of variables , xor and xnor give same ans right ?

@Arjun sir same doubt. :(

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@tusharp assume option(b) as ${X}$ then option(a) becomes ${X}'$, how can they both be equal... option(a) would have been true if (Xnor) was used in between the symbols.

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P xnor Q xnor R = P xor Q xor R ...this is correct (for even variables, xor and xnor are complement of each other, for odd variables, xor and nor are equal)

But option A is (P xor Q xor R)’ and (P xor Q xor R)’ is obviously not equal to (P xor Q xor R)

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$f(P,Q,R)=\sum(1,2,4,7)=P\oplus Q\oplus R$

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@Arjun sir but if 3 variable function is there then exor and exnor are same therefore A also be the ans for this question

 

Answer 2

Draw the k-map, and the pattern like this is always of XOR therefore, the answer will be P⊕Q⊕R

Answer 3

If P=0 and Q=0 our output must be R right
see only B and C gives R
if P=0 and Q=1 our output must be R' right
only B gives R'
hence B is the answer