topic - solutions adda

Question 1

Consider the Boolean operator with the following properties:

Then x#y is equivalent to

A
B
C
D

Digital-Logic-Design Boolean-Algebra GATE 2016 [Set-1]

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

The number of min-terms after minimizing the following Boolean expression is ______.

      [D′ + AB′ + A′C + AC′D + A′C′D]′

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

Consider the following Boolean expression for F:

   F(P, Q, R, S) = PQ + P'QR + P'QR'S

The minimal sum-of-products form of F is

A
B
C
D

Digital-Logic-Design Boolean-Algebra GATE 2014 [Set-1]

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

The truth table

represents the Boolean function

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

The simplified SOP (sum of product) form of the boolean expression

A
B
C
D

Digital-Logic-Design Boolean-Algebra GATE 2011

Question 6 Explanation:

Question 7

The minterm expansion of is

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 GATE 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

If P, Q, R are Boolean variables, then

Simplifies to

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	2^2ⁿ
D	2^n²

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 2^2ⁿ.
Formula: The number of m-ary functions possible with n k-ary variables is m^kⁿ.

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

The simultaneous equations on the Boolean variables x, y, z and w,

      x + y + z = 1
             xy = 0
         xz + w = 1
      xy +  = 0

have the following solution for x, y, z and w, respectively.

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

What values of A, B, C and D satisfy the following simultaneous Boolean equations?

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.

Question 13

Any set of Boolean operators that is sufficient to represent all Boolean expressions is said to be complete. Which of the following is not complete?

A	{ AND, OR }
B	{ AND, NOT }
C	{ NOT, OR }
D	{ NOR }

Digital-Logic-Design Boolean-Algebra ISRO-2018

Question 13 Explanation:

→ NOT, AND and OR gates are basic gates
→ NAND and NOR gates are universal gates.
→ With the help of universal gates, we can construct any boolean expressions. These gates are also called functionally complete.
→ AND+NOT=NAND→ Functionally Complete
→ OR+NOT=NOR→ Functionally Complete
→ NOR→ Functionally Complete
→ AND+OR→ Not functionally complete

Question 14

The Boolean theorem AB + A’C + BC = AB + A’C corresponds to

A	(A + B) ∙ (A’ + C) ∙ (B + C) = (A + B) ∙ (A’ + C)
B	AB + A’C + BC = AB + BC
C	AB + A’C + BC = (A + B) ∙ ( A ‘+ C) ∙ (B + C)
D	(A + B) ∙ (A’ + C) ∙ (B + C) = AB + A’C

Digital-Logic-Design Boolean-algebra ISRO CS 2008

Question 14 Explanation:

(X+Y)*(X+Z)*(Y+Z)=(X+Y)*(X+Z) Consensus Law
XY+XZ+YZ=XY+XZ Consensus Law

For min-term: AB + A'C +BC = AB + A'C
AND for max-term : ( A + B ).( A' + C ).( B + C ) = ( A + B ).(A' + C )

Question 15

In the given network of AND and OR gates, f can be written as

A	X₀X₁X₂ … X_n + X₁X₂ … X_n + X₂X₃ … X_n + ⋯ + X_n
B	X₀X₁ + X₂X₃ + … X_n-1 X_n
C	X₀ + X₁ + X₂ + … + X_n
D	X₀X₁ + X₃ … X_n−1 + X₂X₃ + X₅ … X_n−1 + ⋯ + X_n−2X_n−1 + X_n
E	None of the above

Digital-Logic-Design Boolean-algebra ISRO CS 2008

Question 15 Explanation:

(X₀X₁+X₂)X₃+X₄)X₅+⋯+X_N
=(X₀X₁X₃+X₂X₃+X₄)X₅+⋯+X_N

=X₀X₁X₃X₅+X₂X₃X₅+X₄X₅+⋯+X_N

=X₀X₁X₃X₅⋯X_N−1+X₂X₃X₅⋯X_N−1+X₄X₅X₇⋯X_N−1+⋯+X_N

Question 16

The Boolean expression ( A + C’)(B’+ C’) simplifies to

A	C’ + AB’
B	C’ (A’ + B)
C	B’C’ + AB’
D	None of these.

Digital-Logic-Design Boolean-algebra ISRO CS 2008

Question 16 Explanation:

The following expression can be simplified as:

(A + C')(B'+ C')

= AB' + AC' + B'C' + C
'
= AB' + C'(A + B' + 1) // 1 + A = 1

= AB' + C'

Question 17

In the expression A'(A’ + B’) by writing the first term A as A + 0, the expression is best simplified as

A	A + AB
B	AB
C	A'
D	A + B

Digital-Logic-Design Boolean-Algebra ISRO CS 2008

Question 17 Explanation:

A'(A’ + B’)

This expression can be simplified as:

= A'A' + A'B'

= A' + A'B'

= A'(1 + B') // 1 + B' = 1

= A'

Question 18

Which of the following is not a valid rule of XOR?

A	0 XOR 0 = 0
B	1 XOR 1 = 1
C	1 XOR 0 = 1
D	B XOR B = 0

Digital-Logic-Design Boolean-Algebra ISRO CS 2008

Question 18 Explanation:

XOR gate only returns 1 as the output when both inputs are different and in every other case, it returns 0.

So, options (A), (C) and (D) are correct.

Question 19

In Boolean algebra, rule (X+Y)(X+Z) =

A	Y+XZ
B	X+YZ
C	XY+Z
D	XZ+Y

Digital-Logic-Design Boolean-Algebra ISRO CS 2011

Question 19 Explanation:

in Boolean algebra,
(X+Y)(X+Z) = X + XZ + XY + YZ
= X(1 + Z + Y) + YZ // as (1 + A = A)
= X.1 + YZ
= X + YZ

Question 20

Which logic gate is used to detect overflow in 2’s complement arithmetic?

A	OR gate
B	AND gate
C	NAND gate
D	XOR gate

Digital-Logic-Design Boolean-algebra ISRO CS 2013

Question 20 Explanation:

Question 21

Which of the following is not valid Boolean algebra rule?

A	X.X = X
B	(X + Y).X = X
C	X̄ + XY = Y
D	(X + Y).(X + Z) = X + YZ

Digital-Logic-Design Boolean-Algebra ISRO CS 2014

Question 21 Explanation:

i) X.X = X
ii) (X + Y).X
= X.X + X.Y
= X + X.Y
= X(1 + Y)
= X
iii) X' + XY
= (X' + X)(X' + Y)
= (1)(X' + Y)
= (X' + Y)
iv) (X + Y).(X + Z)
= X.X + X.Z + X.Y + Y.Z
= X(1 + Z + Y) + Y.Z
= X + Y.Z

Question 22

Perform the following operation for the binary equivalent of the decimal numbers

(-14)₁₀ + (-15)₁₀

The solution in 8 bit representation is :

A	11100011
B	00011101
C	10011101
D	11110011

Digital-Logic-Design Boolean-Algebra UGC-NET CS 2018 JUNE Paper-2

Question 22 Explanation:

(-14)₁₀ + (-15)₁₀ = (-29)₁₀

(29)₁₀ = (11101)₂
(29)₁₀ in 8-bit representation = (00011101)₂

(-29)₁₀ = (00011101)₂

Question 23

In digital logic, if A ⊕ B=C, then which one of the following is true?

A	A ⊕ C=B
B	B ⊕ C=A
C	A ⊕ B ⊕ C=0
D	Both A) and B)

Digital-Logic-Design Boolean-Algebra Nielit Scientist-B CS 22-07-2017

Question 23 Explanation:

Question 24

Which of the following logic expression is incorrect?

A	1 ⊕ 0=1
B	1 ⊕ 1 ⊕ 0=1
C	1 ⊕ 1 ⊕ 1=1
D	1 ⊕ 1 =0

Digital-Logic-Design Boolean-Algebra Nielit Scientist-B CS 2016 march

Question 24 Explanation:

This ⊕ symbol is nothing but Ex-OR

Option A: Here, 1 ⊕ 0=1 true according to truth table.
Option B: 1⊕1=0⊕0=0 false
Option C: 1 ⊕ 1=0 ⊕ 1=1 true
Option D: 1 ⊕ 1 =0 true.

Question 25

The boolean expression A'⋅B + A⋅B' + A⋅B is equivalent to

A	A+B
B	A⋅B
C	(A+B)'
D	A'⋅B

Digital-Logic-Design Boolean-Algebra UGC-NET CS 2018 DEC Paper-2

Question 25 Explanation:

Question 26

The relation ≤ and < on a boolean algebra are defined as :

x ≤ y and only if x ∨ y = y
x < y means x ≤ y but x ≠ y
x ≥ y means y ≤ x and
x > y means y < x

Consider the above definitions, which of the following is not true in the boolean algebra ?

(i)

If x ≤ y and y ≤ z, then x ≤ z

(ii)

If x ≤ y and y ≤ x, then x = y

(iii)

If x < y and y < z, then x ≤ y

(iv)

If x < y and y < z, then x < y

Choose the correct answer from the code given below:

Code:

A	(iv) only
B	(iii) only
C	(i) and (ii) only
D	(ii) and (iii) only

Digital-Logic-Design Boolean-Algebra UGC-NET CS 2018 DEC Paper-2

Question 26 Explanation:

iii) “If x < y and y < z, then x ≤ y” is not true.
Because x < y means x ≤ y but x ≠ y.
ii) If x ≤ y and y ≤ x, then x = y is true
Because
x ≤ y implies x v y = x
y ≤ x implies x v y = y
x v y = x = y

Question 27

Consider the following boolean equations :

(i) wx + w(x + y) + x(x + y)= x + wy
(ii) (wx’(y + xz’) + w’x’)y = x’y

What can you say about the above equations ?

A	Both (i) and (ii) are true
B	(i) is true and (ii) is false
C	Both (i) and (ii) are false
D	(i) is false and (ii) is true

Digital-Logic-Design Boolean-Algebra UGC-NET CS 2018 DEC Paper-2

Question 27 Explanation:

(i)
wx + w(x + y) + x(x + y)
= (wx + wx) + wy + (x + xy)
= wx + wy + x(1 + y)
= wx + wy + x
= (w + 1)x + wy
= x + wy
(ii)

Question 28

Find the boolean expression for the logic circuit shown below :

(1-NAND gate, 2-NOR gate, 3-NOR gate)

A	AB
B	AB’
C	A’B’
D	A’B

Digital-Logic-Design Boolean-Algebra UGC-NET CS 2018 DEC Paper-2

Question 28 Explanation:

Question 29

Which of the following statements are true ?

(i)

Every logic network is equivalent to one using just NAND gates or just NOR gates.

(ii)

Boolean expressions and logic networks correspond to labelled acyclic digraphs.

(iii)

No two Boolean algebras with n atoms are isomorphic.

(iv)

Non-zero elements of finite Boolean algebras are not uniquely expressible as joins of atoms.

Choose the correct answers from the code given below :

Code :

A	(i) and (iv) only
B	(i) and (ii) only
C	(i), (ii) and (iii) only
D	(ii), (iii) and (iv) only

Digital-Logic-Design Boolean-Algebra UGC-NET CS 2018 DEC Paper-2

Question 29 Explanation:

→ A universal logic gate is a logic gate that can be used to construct all other logic gates.
→ NAND and NOR are universal gates, by using these gates we can construct all gates.
→ An atom of a Boolean algebra is an element x such that there exist exactly two elements y satisfying y ≤ x, namely x and 0. A Boolean algebra is said to be atomic when every element is a sup of some set of atoms (the bottom element is always the empty sup).
→ So, the options (iii) and (iv) are false.

Question 30

In digital logic, if A ⊕ B=C, then which one of the following is true?

A	A ⊕ C=B
B	B ⊕ C=A
C	A ⊕ B ⊕ C=0
D	Both A) and B)
E	None of these

Digital-Logic-Design Boolean-Algebra Nielit Scientist-B IT 22-07-2017

Question 30 Explanation:

Given that
XOR:
X ⊕ X= 0
X ⊕ X'= 1
X ⊕ 0 = X
X ⊕ 1 = X'
A) A ⊕ C = B
A ⊕ A ⊕ B = B
0 ⊕ B = B
B= B
B) B ⊕C = A
B ⊕ A ⊕ B = A
A ⊕ 0=A
A=A
C) A⊕B=C
A⊕B⊕ C=C ⊕C ,
A⊕B⊕ C= 0
All are correct

Question 31

Which of the following does NOT represent the Exclusive NOR operation over the binary variables A and B?

A	A’ ⊕ B’
B	A ⊕ B’
C	A’ ⊕ B
D	AB + A’B’

Digital-Logic-Design Boolean-Algebra JT(IT) 2018 PART-B Computer Science

Question 31 Explanation:

A’ ⊕ B’ = Ex-NOR

Question 32

The term sum of product in boolean algebra means

A	The AND function of several AND functions
B	The AND function of several OR functions
C	The OR function of several AND functions
D	The OR function of several OR functions

Digital-Logic-Design Boolean-Algebra KVS DEC-2017

Question 32 Explanation:

Sum of product: A boolean expression consisting purely of Minterms (product terms) is said to be in canonical sum of products form.
Example: F=AB+AC+AD Product of sum: A boolean expression consisting purely of Maxterms (sum terms) is said to be in canonical product of sums form.
Example: F=(A+B).(A+C).(A+D)

Question 33

In boolean algebra, (x ⋀ y)’ = x’ V y’ and (x V y)’ = x’ ⋀ y’ is known as ___ law.

A	Demorgan’s law
B	Absorption
C	Dominance
D	Idempotent

Digital-Logic-Design Boolean-Algebra JT(IT) 2016 PART-B Computer Science

Question 33 Explanation:

The rules can be expressed in English as:
1. The negation of a disjunction is the conjunction of the negations.
2. The negation of a conjunction is the disjunction of the negations.

Question 34

The boolean expression (A+C)(AB'+AC)(AC'+B') can be simplified as

A	A'B+BC
B	AB'
C	AB+BC
D	AB+A'C

Digital-Logic-Design Boolean-Algebra KVS DEC-2017

Question 34 Explanation:

(A+C)(AB'+AC)(AC'+B')
First compute one and primes
(AAB'+AAC+AB'C+ACC)(AC'+B')
to take common AC then we get
(AB'+AC+AB'C+AC)(AC'+B')
To take common AB'
(AB'(1+C)+AC)(AC'+B')
(AB'+AC)(AC'+B')
AB'AC'+AB'B'+ACAC'+ACB'
AB'C'+AB'+0+ACB'
AB'(C'+1)+ACB'
AB'+ACB'
AB'(1+C)
AB'
(or)
We can also use truth table to get same solution.

Question 35

How many gates would be required to implement the following boolean expression after simplification? Expression:

A	4
B	6
C	8
D	10

Digital-Logic-Design Boolean-Algebra KVS 30-12-2018 Part B

Question 35 Explanation:

Given expression is =A’+B’+C’+A’C+B’C+A’BC’+AB’+AC’+AB’C
=A’+A’C+B’+B’C+C’+AC’+A’BC’+AB’+AB’C
=A’ (1+C)+B’(1+C)+C’(1+A)+A’BC’+AB’(1+C) [ 1+X=1]
=A’+B’+C’+A’BC’+AB’
=A’+C’+A’BC’+B’ (1+A)
=A’+C’+A’BC’+B’
=A’+B’+C’ (1+A’B)
=A’+B’+C’
Total four gates required three not gates and one OR gate.

Question 36

A boolean operator θ is defined as follows:
1θ1=1
1θ0=0
0θ1=0
0θ0=1
What will be the truth value of the expression (XθY)θZ=Xθ(YθZ)?

A	Always false
B	Always true
C	Sometimes true
D	Sometimes false

Digital-Logic-Design Boolean-Algebra KVS 30-12-2018 Part B

Question 36 Explanation:

(XθY)θZ=Xθ(YθZ) is always true if θ is associative.
From the given table, we can infer that θ is Ex-NOR operator.
Ex-NOR is associative. Hence (XθY)θZ=Xθ(YθZ) is always true

Question 37

Consider the graph given below :

Use Kruskal’s algorithm to find a minimal spanning tree for the graph. The List of the edges of the tree in the order in which they are chosen is?

A	AD, AE, AG, GC, GB, BF
B	GC, GB, BF, GA, AD, AE
C	GC, AD, GB, GA, BF, AE
D	AD, AG, GC, AE, GB, BF

Digital-Logic-Design Boolean-Algebra UGC NET CS 2017 Nov- paper-2

Question 37 Explanation:

Kruskal’s algorithm will sort all edge weight first and according to spanning tree rules it construct Minimum cost spanning tree.
Here, some edges have same weight. This situation, we can get many number of spanning trees.
We have to connect strictly below order.
Note: DB not given edge weight. GC, AD, BG, AE, BF
Edge weight 2= AD or GC
Edge weight 3= AG or GB
Edge weight 4= AE or DG or BC or BF
Edge weight 5= ED or CF
Edge weight 6: AC

Question 38

The Boolean function [~(~p ∧ q) ∧ ~( ~p ∧ ~q)] ∨ (p ∧ r) is equal to the Boolean function:

A	q
B	p ∧ r
C	p ∨ q
D	p

Digital-Logic-Design Boolean-Algebra UGC NET CS 2016 Aug- paper-2

Question 38 Explanation:

Question 39

Which of the following logic expressions is incorrect?

A	1 ⊕ 0 = 1
B	1 ⊕ 1 ⊕ 1 = 1
C	1 ⊕ 1 ⊕ 0 = 1
D	1 ⊕ 1 = 0

Digital-Logic-Design Boolean-Algebra UGC NET CS 2016 July- paper-2

Question 39 Explanation:

Here, ⊕ is nothing but Ex-OR operator. The truth table for Ex-OR is

According to truth table,
Option-A is TRUE
Option-B is a 1 ⊕ 1 is 0.
0 ⊕ 1 is 1(TRUE)
Option-C is 1 ⊕ 1 is 0.
0 ⊕ 0 = 0 but given 1. So, FALSE
Option-D is TRUE.

Question 40

Simplified Boolean equation for the following truth table is:

A	F = yz’ + y’z
B	F = xy’ + x’y
C	F = x’z + xz’
D	F = x’z + xz’ + xyz

Digital-Logic-Design Boolean-Algebra UGC NET CS 2016 July- paper-2

Question 40 Explanation:

Method-1: Using K-Map

Method-2: Using boolean simplification
= x’y’z+x’yz+xy’z’+xyz’
= x'z(y'+y)+ xz'(y'+y)
= x'z+xz' (Since y'+y=1)

Question 41

The simplified form of a Boolean equation (AB’ + AB’C + AC) (A’C’ + B’) is :

A	AB’
B	AB’C
C	A’B
D	ABC

Digital-Logic-Design Boolean-Algebra UGC NET CS 2016 July- paper-2

Question 41 Explanation:

(AB’ + AB’C + AC) (A’C’ + B’)
= (AB'+AC) (A'C'+B')
= AB'A'C' + AB'B' + ACA'C' + ACB'
= AB'B' + ACB'
= AB'(C+1)
= AB'

Question 42

The boolean expression A’⋅B+A.B’+A.B is equivalent to

A	A+B
B	A.B
C	(A+B)’
D	A’.B

Digital-Logic-Design Boolean-aLGEBRA UGC NET CS 2018-DEC Paper-2

Question 42 Explanation:

Question 43

The relation ≤ and < on a boolean algebra are defined as :
x ≤ y and only if x ∨ y = y
x < y means x ≤ y but x ≠ y
x ≥ y means y ≤ x and
x > y means y <x
Consider the above definitions, which of the following is not true in the boolean algebra ?
(i)If x ≤ y and y ≤ z, then x ≤ z
(ii)If x ≤ y and y ≤ x, then x=y
(iii)If x < y and y < z, then x ≤ y
(iv)If x < y and y < z, then x < y

A	(iv) only
B	(iii) only
C	(i) and (ii) only
D	(ii) and (iii) only

Digital-Logic-Design Boolean-Algebra UGC NET CS 2018-DEC Paper-2

Question 43 Explanation:

iii) “If x < y and y < z, then x ≤ y” is not true.
Because x < y means x ≤ y but x ≠ y.
ii)If x ≤ y and y ≤ x, then x=y is true
Because
x ≤ y implies x v y =x
y ≤ x implies x v y = y
X v y = x = y
Note: From the given definitions, x In option 3, the condition x≠y is missing.
Key point: y < z has nothing to do with x ≤ y. So, we are ignoring.

Question 44

Which of the following statements are true ?
(i) Every logic network is equivalent to one using just NAND gates or just NOR gates.
(ii) Boolean expressions and logic networks correspond to labelled acyclic digraphs.
(iii) No two Boolean algebras with n atoms are isomorphic.
(iv) Non-zero elements of finite Boolean algebras are not uniquely expressible as joins of atoms.

A	(i) and (iv) only
B	(i) and (ii) only
C	(i), (ii) and (iii) only
D	(ii), (iii) and (iv) only

Digital-Logic-Design Boolean-Algebra UGC NET CS 2018-DEC Paper-2

Question 44 Explanation:

→ A universal logic gate is a logic gate that can be used to construct all other logic gates.
→ NAND and NOR are universal gates, by using these gates we can construct all gates.
→ An atom of a Boolean algebra is an element x such that there exist exactly two elements y satisfying y ≤ x, namely x and 0. A Boolean algebra is said to be atomic when every element is a sup of some set of atoms (the bottom element is always the empty sup).
→ So the options (iii) and (iv) are false

Question 45

Match the following identities/laws to their corresponding name :

A	a-iii, b-iv, c-i, d-ii
B	a-iv, b-iii, c-i, d-ii
C	a-iv, b-iii, c-ii, d-i
D	a-iii, b-iv, c-ii, d-i

Digital-Logic-Design Boolean-Algebra UGC NET CS 2013 Sep-paper-2

Question 45 Explanation:

Idempotent laws: x+x=x and x.x=x
Identity laws: x+0=x and x.1=x
Dominance laws: x+1=1 and x.0=0
Absorption laws: x.(x+y)=x

Question 46

The absorption law in Boolean algebra say that

A	X + X = X
B	X . X = X
C	x + x . y = x
D	None of the above

Digital-Logic-Design Boolean-Algebra UGC NET CS 2011 June-Paper-2

Question 46 Explanation:

Absorption laws in Boolean algebra:
X • (X+Y) = X
X + X•Y = X

Question 47

The idempotent law in Boolean algebra says that :

A	~(~x)=x
B	x+x=x
C	x+xy=x
D	x(x+y)=x

Digital-Logic-Design Boolean-Algebra UGC NET CS 2008-june-Paper-2

Question 47 Explanation:

→ Idempotent law:
The idempotence in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from idem + potence (same + power).
(i). X + X=X
(ii). X ∧ X=X
According to boolean algebra

Question 48

If p and q are two statements and they take truth values p = 1 and q = 1, then their conjunction p and q written as p ^ q takes truth value :

A	0
B	1
C	-1
D	None

Digital-Logic-Design Boolean-Algebra TNPSC-2017-Polytechnic-CS

Question 48 Explanation: