Digital Logic & Number representation

Last Updated : 27 Sep, 2021

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Question 1

In the following truth table, V = 1 if and only if the input is valid.

What function does the truth table represent?

A	Priority encoder
B	Decoder
C	Multiplexer
D	Demultiplexer

GATE CS 2013 Digital Logic & Number representation Combinational Circuits
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Question 1 Explanation:

Since there are more than one outputs and number of outputs is less than inputs, it is a Priority encoder V=1 when input is valid and for priority encoder it checks first high bit encountered. Except all are having at least one bit high and ‘x’ represents the “don’t care” as we have found a high bit already. So answer is (A).

Question 2

Which one of the following expressions does NOT represent exclusive NOR of x and y?

A	xy+x'y'
B	x⊕y'
C	x'⊕y
D	x'⊕y'

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Question 2 Explanation:

By Definition of XNOR, $x\odot y = x' y' + xy$ So Option-A is correct. Also by Definition of XOR, $x\oplus y = x' y + xy'$ Option-B is $x\oplus y' = x' y' + x(y')' = x' y' + xy = x\odot y$ So Option-B is also correct. Option-C is $x' \oplus y = (x')' y + x' y' = x' y' + xy = x\odot y$ Option-C is also correct. Option-D x'⊕y' = x''y' + x'y'' = xy' + x'y = x⊕y ≠ x⊙y Therefore option (D) is false. This explanation is provided by Chirag Manwani.

Question 3

The truth table

represents the Boolean function

A	X
B	X+Y
C	X xor Y
D	Y

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Question 3 Explanation:

The value of f(X, Y) is same as X for all input pairs. Also sum of product form of expression we get,

= XY’+XY
= X(Y’+Y)
= X *1
= X

We see from truth table –

Column x = f(x,y) 
So , 
f(x,y)=x

Option (A) is correct.

Question 4

What is the minimal form of the Karnaugh map shown below? Assume that X denotes a don’t care term.

A	b'd'
B	b'd' + b'c'
C	b'd' + a'b'c'd'
D	b'd' + b'c' + c'd'

Digital Logic & Number representation Digital Logic & Number representation Logic functions and Minimization
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Question 4 Explanation:

There are two prime implicants in the following K-Map-

Prime Implicant higlighted in Green = $b'c'$ Prime Implicant higlighted in Orange = $b'd'$ So the Boolean expression is- $b'c' + b'd'$ Therefore option (B) is correct. This explanation is provided by Chirag Manwani.

Question 5

What is the minimal form of the Karnaugh map shown below? Assume that X denotes a don’t care term.

A	b'd'
B	b'd' + b'c'
C	b'd' + a'b'c'd'
D	b'd' + b'c' + c'd'

Digital Logic & Number representation Digital Logic & Number representation Logic functions and Minimization
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Question 5 Explanation:

There are two prime implicants in the following K-Map-

Question 6

Which one of the following circuits is NOT equivalent to a 2-input XNOR (exclusive NOR) gate?

A	A
B	B
C	C
D	D

GATE CS 2011 Digital Logic & Number representation Logic functions and Minimization
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Question 6 Explanation:

All options except D produce XOR as described below : $NOT({a \bigoplus b}) = XNOR \\ NOT(NOT(a)\bigoplus NOT(b)) = NOT(aNOT(b) + NOT(a)b) = XNOR \\ NOT(a) \bigoplus b = ab + NOT(ab)= XNOR \\ NOT(NOT(aNOT(b))(NOT(b)+a)) = NOT(NOT(ab)+ba) = NOT(XNOR)$

Question 7

The simplified SOP (Sum Of Product) form of the boolean expression (P + Q' + R') . (P + Q' + R) . (P + Q + R') is

A	(P'.Q + R')
B	(P + Q'.R')
C	(P'.Q + R)
D	(P.Q + R)

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Question 7 Explanation:

See following : (P+Q'+R').(P+Q'+R).(P+Q+R') = $\prod(3, 2, 1) = \sum(0, 4, 5, 6, 7)$ gate2011A24

From the K-map, POS form is : P + Q'.R'

Question 8

Consider the following circuit involving three D-type flip-flops used in a certain type of counter configuration.

If at some instance prior to the occurrence of the clock edge, P, Q and R have a value 0, 1 and 0 respectively, what shall be the value of PQR after the clock edge?

A	000
B	001
C	010
D	011

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Question 8 Explanation:

P' = R Q' = (P + R)' R' = QR' Given that (P, Q, R) = (0, 1, 0), next state P', Q', R' = 0, 1, 1 ----------------------------------------------------------------------------------------------- D flip flop truth table

D	Q(t+1)
0	0
1	1

Initially (p,q,r) =(0,1,0) D for p=R D for q=NOT(p xor r) D for r= (not)r.q So Q(t+1) for(p,q,r) p=>r=0 so p=0 q=> NOT(p xor r) => 1 so q=1 r=>(not)r.q => 1 so r=1 (p, q, r) = (0, 1, 1) Alternative approach - Truth table of a D Flip-Flop- $\begin{tabular}{|c|c|} \hline D_{t} & Q_{t+1}\\ \hline 0 & 0\\ \hline 1 & 1\\ \hline \end{tabular}$ By looking at the circuit diagram, it is clear that the boolean expressions of P, Q, and R are- Here the subscript t refers to the current clock cycle, and the subscript (t+1) refers to the next clock cycle. $Q_{P(t+1)} = P_{t+1} = R_t$ $Q_{Q(t+1)} = Q_{t+1} = R_{t}' P_{t}'$ $Q_{R(t+1)} = R_{t+1} = Q_{t} R_{t}'$ $\begin{tabular}{|c|c|c|} \hline D_{t} & Q_{t+1}\\ \hline 0 & 0\\ \hline 1 & 1\\ \hline \end{tabular}$ This explanation is provided by Chirag Manwani.

Question 9

Consider the data given in previous question. If all the flip-flops were reset to 0 at power on, what is the total number of distinct outputs (states) represented by PQR generated by the counter?

A	3
B	4
C	5
D	6

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Question 9 Explanation:

There are four distinct states, 000 → 010 → 011 → 100 (→ 000) so the answer is B

Question 10

The minterm expansion of f(P, Q, R) = PQ + QR' + PR' is

A	m2 + m4 + m6 + m7
B	m0 + m1 + m3 + m5
C	m0 + m1 + m6 + m7
D	m2 + m3 + m4 + m5

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Question 10 Explanation:

K-map,

= PQ + QR’ + PR’ 
= PQ(R+R’) + (P+P’)QR’ + P(Q+Q’)R’
= PQR + PQR’ +PQR’ +P’QR’ + PQR’ + PQ’R’ 
= PQR(m7) + PQR'(m6)+P’QR'(m2) +PQ’R'(m4) 
= m2 + m4 + m6 + m7

Option (A) is correct.

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Digital Logic & Number representation

Digital Logic & Number representation

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