Boolean-Algebra

Question 1

Consider the Boolean operator with the following properties:

Then x#y is equiavlent to

   
A
B
C
D
       Digital-Logic-Design       Boolean-Algebra       2016 set-01
Question 1 Explanation: 


Ex-OR satisfies all the properties. Hence,
Question 2

Let, x1⊕x2⊕x3⊕x4 = 0 where x1, x2, x3, x4 are Boolean variables, and ⊕ is the XOR operator. Which one of the following must always be TRUE?

A
x1x2x3x4 = 0
B
x1x3+x2 = 0
C
D
x1 + x2 + x3 + x4 = 0
       Digital-Logic-Design       Boolean-Algebra       GATE 2016 set-2
Question 2 Explanation: 
Given expression is,
x1 ⊕ x2 ⊕ x3 ⊕ x4 = 0 -----(1)
A) x1x2x3 x4 = 0
Put x1 = 1, x2 = 1, x3 = 1, x4 = 1
The given equation will be zero, i.e.,
1 ⊕ 1 ⊕ 1 ⊕ 1 = 0
But,
x1x2x3 x4 ≠ 0
So, false.
B) x1x3 + x2 = 0
Put x1 = 1, x2 = 1, x3 = 0 , x4 = 0
The given equation will be zero, i.e.,
1 ⊕ 1 ⊕ 0 ⊕ 0 = 0
But,
x1x3 + x2 ≠ 0
So, false.
D) x1 + x2 + x3 + x4 = 0
Let x1=1, x2=1, x3=0, x4=0
The given equation will be zero, i.e.,
1 ⊕ 1 ⊕ 0 ⊕ 0 = 0
But,
x1 + x2 + x3 + x4 ≠ 0
So, false.
(i) True.
Question 3

A
1
B
2
C
3
D
4
       Digital-Logic-Design       Boolean-Algebra       GATE 2015 -(Set-2)
Question 3 Explanation: 
Lets simplify it
[D' + AB' + A'C + AC'D + A'C'D]'
[D' + AB' + A'C + C'D (A + A')']' (since A+A' = 1)
[AB' + A'C + (D' + C') (D' + D)]' (since D' + D =1)
[AB' + A'C + D' + C']'
[AB' + (A' + C') (C + C') + D']'
[AB' + A' + C' + D']'
[(A + A') (A' + B') + C' + D']'
[A' + B' + C' + D']'
Apply de-morgan's law,
ABCD
Question 4
 
A
B
C
D
       Digital-Logic-Design       Boolean-Algebra       GATE 2014(Set-01)
Question 4 Explanation: 
PQ + P’QR + P’QR’S
= Q(P+P’R) + P’QR’S
= Q(P+R) + P’QR’S
=QP + QR + P’QR’S
= QP + Q(R + P’R’S)
= QP + Q( R + P’S)
= QP + QR + QP’S
= Q(P+P’S) + QR
= Q(P+S)+ QR
= QP + QS + QR
Question 5
A
X
B
X + Y
C
X ⊕ Y
D
Y
       Digital-Logic-Design       Boolean-Algebra       Gate 2012
Question 5 Explanation: 
f(X,Y)=XY’ + XY = X(Y’ + Y) = X
Question 6
       
A
B
C
D
       Digital-Logic-Design       Boolean-Algebra       Gate 2011
Question 6 Explanation: 
Question 7
     
A
m2+m4+m6+m7
B
m0+m1+m3+m5
C
m0+m1+m6+m7
D
m2+m3+m4+m5
       Digital-Logic-Design       Boolean-Algebra       2010
Question 7 Explanation: 
Convert PQ + QR' + PR' into canonical form
= PQR + PQR' + PQR' + P'QR' + PQR' + PQ'R'
= PQR + PQR' + P'QR' + PQ'R'
=m7 + m6 + m2 + m4
Question 8
 
A
B
C
D
       Digital-Logic-Design       Boolean-Algebra       Gate-2008
Question 8 Explanation: 
Question 9
What is the maximum number of different Boolean functions involving n Boolean variables?
A
n2
B
2n
C
22n
D
2n2
       Digital-Logic-Design       Boolean-Algebra       Gate-2007
Question 9 Explanation: 
Each “boolean” variable has two possible values i.e 0 and 1.
Number of variables= n
Number of input combinations is 2n.
Each “boolean” function has two possible outputs i.e 0 and 1.
Number of boolean functions possible is 22n.
Formula: The number of m-ary functions possible with n k-ary variables is mkn.
Question 10
The Boolean function x'y' + xy + x'y is equivalent to
A
x' + y'
B
x + y
C
x + y'
D
x' + y
       Digital-Logic-Design       Boolean-Algebra       Gate-2004
Question 10 Explanation: 
x'y' + xy + x'y
= x'y' + x'y + xy
= x'(y'+y)+xy
= x'⋅1+xy
= x'+xy
= (x'+x)(x'+y)
= 1⋅(x'+y)
= x'+y
Question 11
       
A
0 1 0 0
B
1 1 0 1
C
1 0 1 1
D
1 0 0 0
       Digital-Logic-Design       Boolean-Algebra       Gate-2000
Question 11 Explanation: 
Just put the values of each options in the equation and check it.
Question 12
   
A
A = 1, B = 0, C = 0, D = 1
B
A = 1, B = 1, C = 0, D = 0
C
A = 1, B = 0, C = 1, D = 1
D
A = 1, B = 0, C = 0, D = 0
       Digital-Logic-Design       Boolean-Algebra       Gate-1995
Question 12 Explanation: 
For verification, just put up the values and check for AND, OR operations and their outputs.
There are 12 questions to complete.