# Digital Logic & Number representation

Question 1 |

In the following truth table, V = 1 if and only if the input is valid.
What function does the truth table represent?

Priority encoder | |

Decoder | |

Multiplexer | |

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?

xy+x'y' | |

x⊕y' | |

x'⊕y | |

x'⊕y' |

**GATE CS 2013**

**Digital Logic & Number representation**

**Logic functions and Minimization**

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Question 2 Explanation:

By Definition of XNOR,
So Option-A is correct.
Also by Definition of XOR,
Option-B is
So Option-B is also correct.
Option-C is
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 |

X | |

X+Y | |

X xor Y | |

Y |

**GATE CS 2012**

**Digital Logic & Number representation**

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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 = XWe see from truth table –

Column x = f(x,y) So , f(x,y)=xOption (A) is correct.

Question 4 |

b'd' | |

b'd' + b'c' | |

b'd' + a'b'c'd' | |

b'd' + b'c' + c'd' |

**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 =
Prime Implicant higlighted in Orange =
So the Boolean expression is-
Therefore option (B) is correct.
This explanation is provided by

**Chirag Manwani**.Question 5 |

b'd' | |

b'd' + b'c' | |

b'd' + a'b'c'd' | |

b'd' + b'c' + c'd' |

**Digital Logic & Number representation**

**Logic functions and Minimization**

**Discuss it**

Question 5 Explanation:

There are two prime implicants in the following K-Map-
Prime Implicant higlighted in Green =
Prime Implicant higlighted in Orange =
So the Boolean expression is-
Therefore option (B) is correct.
This explanation is provided by

**Chirag Manwani**.Question 6 |

A | |

B | |

C | |

D |

**GATE CS 2011**

**Digital Logic & Number representation**

**Logic functions and Minimization**

**Discuss it**

Question 6 Explanation:

All options except D produce XOR as described below :

Question 7 |

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

(P'.Q + R') | |

(P + Q'.R') | |

(P'.Q + R) | |

(P.Q + R) |

**GATE CS 2011**

**Digital Logic & Number representation**

**Logic functions and Minimization**

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Question 7 Explanation:

See following :
(P+Q'+R').(P+Q'+R).(P+Q+R') =
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?

000 | |

001 | |

010 | |

011 |

**GATE CS 2011**

**Digital Logic & Number representation**

**Sequential circuits**

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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

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)

D | Q(t+1) |

0 | 0 |

1 | 1 |

**Alternative approach -**Truth table of a D Flip-Flop- 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. 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?

3 | |

4 | |

5 | |

6 |

**GATE CS 2011**

**Digital Logic & Number representation**

**Sequential circuits**

**Discuss it**

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

m2 + m4 + m6 + m7 | |

m0 + m1 + m3 + m5 | |

m0 + m1 + m6 + m7 | |

m2 + m3 + m4 + m5 |

**GATE CS 2010**

**Digital Logic & Number representation**

**Logic functions and Minimization**

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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 + m7Option (A) is correct.

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