Discrete Structures Subject Wise UGC NET Question Analysis Part-1

Q1➡ | NET June 2021
Let (X,* ) be a semi-group. Furthermore, for every a and b in X, if a ≠ b, then a*b ≠ b*a. Based on the defined semi-group,
choose the correct equalities from the options given below:
A. For every a in X, a*a = a
B. For every a, b in X, a*b *a= a
C. For every a, b, c in X, a*b *c= a*c
i ➥ A and B only
ii ➥ A and C only
iii ➥ A, B and C
iv ➥ B and C only
Answer: III

Explanation:
Given,
(X,* ) be a semi-group.
for every a and b in X, if a ≠ b, then a*b ≠ b*a.

Concept,
Semi Group(X,* ) have 2 properties, which are:
1) Closure Property
if a and b in X, then a*b∈X

2) Associative Property
if a,b and c in X, then
(a*b)*c = a*(b*c)

Let’s solve,
A) For every a in X, a*a = a

Lets assume a*a ≠ a

Lets suppose a*a = b, where a ≠ b
⟹ a*(a*a) = a*b
⟹ (a*a)*a = a*b { by associative property, a*(b*c) = (a*b)*c}
⟹ b*a = a*b { we suppose a*a = b}

Since it is given, for every a and b in X, if a ≠ b, then a*b ≠ b*a, but here a*b = b*a, which proves that our initial assumption is wrong.
Hence, For every a in X, a*a = a (Correct)

B) For every a, b in X, a*b *a= a

Lets assume a*b*a ≠ a

Lets suppose a*b*a = b, where a ≠ b

it is given, a*b ≠ b*a, and we already suppose a*b*a = b. Now put b = a*b*a in a*b ≠ b*a
⟹ a*(a*b*a) ≠ (a*b*a)*a
⟹ (a*a)*b*a ≠ a*b*(a*a) { by associative property, a*(b*c) = (a*b)*c}
⟹ a*b*a ≠ a*b*a { it is given in A, a*a = a and we already proved that this is correct}

Here, a*b*a ≠ a*b*a but it is actually equal, which proves that our initial assumption is wrong.
Hence, For every a, b in X, a*b *a= a (Correct)

C) For every a, b, c in X, a*b *c= a*c

Lets assume a*b*c ≠ a*c

Lets suppose a*b*c = a, and a*c=b where a ≠ b

it is given, a*b ≠ b*a, and we already suppose a*b*c = a and a*c=b . Now put a= a*b*c and b = a*c in a*b ≠ b*a
⟹ (a*b*c)*(a*c) ≠ (a*c)*(a*b*c)
⟹ a*b*(c*a*c) ≠ (a*c*a)*b*c { by associative property, a*(b*c) = (a*b)*c}
⟹ a*b*c ≠ a*b*c { it is given in B, a*b *a= a and we already proved that this is correct. Like statement B, we can write c*a*c=c and a*c*a=a }

Here, a*b*c ≠ a*b*c but it is actually equal, which proves that our initial assumption is wrong.
Hence, For every a, b in X, a*b *a= a (Correct)

Therefore, Option(III) is correct.
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Q2➡ | NET June 2021
How many ways are there to assign 5 different jobs to 4 different employees if every employee is assigned at least 1 job?
i ➥ 1024
ii ➥ 20
iii ➥ 240
iv ➥ 625
Answer: III

Explanation:
Given,
5 different jobs and 4 different employees
every employee is assigned at least 1 job


Let’s solve,
Lets take 4 employees as A,B,C,D

  How many ways are there to assign 5 different jobs to 4 different employees if every employee is assigned at least 1 job?

Total ways = 60 + 60 + 60 + 60 = 240 ways

Therefore, Option(III) is correct.
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Q3➡ | NET June 2021
Consider the sentence below.
There is a country that borders both India and Pakistan.
Which of the following logical expressions express the above sentence correctly when the predicate Country(x)
represents that x is a country and Borders(x, y) represents that the countries x and y share the border?
i ➥ [∃c Country(c)] ⇒ [Border (c, India) ∧ Border (c,Pakistan)]
ii ➥ ∃c Border (Country(c), India ∧ Pakistan)
iii ➥ ∃c Country(c) ∧ Border (c, India) ∧ Border (c,Pakistan)
iv ➥ ∃c Country(c) ⇒ [Border (c, India) ∧ Border (c,Pakistan)]

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Answer: III
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Q4➡ | NET June 2021
Which of the following Graphs is (are) planar?

choose the correct answer from the options given below:
i ➥ A and B only
ii ➥ A only
iii ➥ B and C only
iv ➥ B only
Answer: II
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Q5➡ | NET June 2021
which of the following are logically equivalent ?
A. ¬p→(q→r)and q→(p∨r)
B. (p→q)→r and p→(q→r)
C. (p→q)→(r→s) and (p→r)→(q→s)

Choose the correct answer from the options below:
i ➥ A and B only
ii ➥ A and C only
iii ➥ A only
iv ➥ B and C only

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Answer: III
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Q6➡ | NET June 2021
For which value of n is Wheel graph Wn regular?
i ➥ 2
ii ➥ 3
iii ➥ 4
iv ➥ 5

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Answer: II
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Q7➡ | NET June 2021
Let us assume a person climbing the stairs can take one stair or two stairs at a time.
How many ways can this person climb a flight of eight stairs?
i ➥ 21
ii ➥ 24
iii ➥ 31
iv ➥ 34

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Answer: IV
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Q8➡ | NET June 2021
Next five questions are based on the following passage.
Consider a domain consisting of three Boolean variables Toothache, Cavity, and Catch. The full joint distribution is a 2×2×2 table as shown in the figure below.

The probability of a toothache, given evidence of a cavity, P(toothache | cavity) is ____.
i ➥ 0.216
ii ➥ 0.280
iii ➥ 0.400
iv ➥ 0.600

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Answer: IV
Explanation:
Formula,
Conditional Probability
The probability of a toothache, given evidence of a cavity, P(toothache | cavity) is ____.
Calculation,
The probability of a toothache, given evidence of a cavity,
 The probability of a toothache, given evidence of a cavity, P(toothache | cavity) is ____.
The probability of a toothache, given evidence of a cavity, P(toothache | cavity) is ____.
P(toothache Ո cavity ) = 0.108 + 0.012 = 0.12

The probability of a toothache, given evidence of a cavity, P(toothache | cavity) is ____.
P(cavity ) = 0.108 + 0.012 + 0.072 + 0.008 = 0.2
 The probability of a toothache, given evidence of a cavity, P(toothache | cavity) is ____.
The probability of a toothache, given evidence of a cavity, P(toothache | cavity) is ____.
So,Option(IV) is correct.

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Q9➡ | NET June 2021
Next five questions are based on the following passage.
Consider a domain consisting of three Boolean variables Toothache, Cavity, and Catch. The full joint distribution is a 2×2×2 table as shown in the figure below.

P(cavity U toothache) is________________.
i ➥ 0.120
ii ➥ 0.200
iii ➥ 0.280
iv ➥ 0.600

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Answer: III
Explanation:
Formula,
P(A U B) = P(A) + P(B) – P(A Ո B)
Calculation,
P(cavity U toothache) = P(cavity) + P(toothache) – P(cavity Ո toothache)
The probability of a toothache, given evidence of a cavity, P(toothache | cavity) is ____.
P(cavity ) = 0.108 + 0.012 + 0.072 + 0.008 = 0.2

P(cavity U toothache) is________________.
P(toothache) = 0.108 + 0.012 + 0.016 + 0.064 = 0.2


P(cavity Ո toothache) = 0.108 + 0.012 = 0.12

P(cavity U toothache) = P(cavity) + P(toothache) – P(cavity Ո toothache)
P(cavity U toothache) = 0.2 + 0.2 – 0.12 = 0.28
So, Option(III) is correct.

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Q10➡ | NET June 2021
Next five questions are based on the following passage.
Consider a domain consisting of three Boolean variables Toothache, Cavity, and Catch. The full joint distribution is a 2×2×2 table as shown in the figure below.

The probability for Cavity, given that either Toothache or Catch is true, P(Cavity | toothache U catch) is _______.
i ➥ 0.4615
ii ➥ 0.5384
iii ➥ 0.6000
iv ➥ 0.8000

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Answer: I
Explanation:
Formula,
The probability for Cavity, given that either Toothache or Catch is true, P(Cavity | toothache U catch) is _______.
Calculation,
 The probability for Cavity, given that either Toothache or Catch is true, P(Cavity | toothache U catch) is _______.
The probability for Cavity, given that either Toothache or Catch is true, P(Cavity | toothache U catch) is _______.
P(cavity Ո (toothache U catch)) = 0.108 + 0.012 + 0.072 = 0.192

 The probability for Cavity, given that either Toothache or Catch is true, P(Cavity | toothache U catch) is _______.
P(toothache U cavity) = 0.108 + 0.012 + 0.072 + 0.016 + 0.064 + 0.144 = 0.416

 The probability for Cavity, given that either Toothache or Catch is true, P(Cavity | toothache U catch) is _______.
 The probability for Cavity, given that either Toothache or Catch is true, P(Cavity | toothache U catch) is _______.= 0.4615
So, Option(I) is correct.

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` Q11➡ | NET June 2021
Next five questions are based on the following passage.
Consider a domain consisting of three Boolean variables Toothache, Cavity, and Catch. The full joint distribution is a 2×2×2 table as shown in the figure below.

The marginal probability of cavity P(cavity) is _____.
i ➥ 0.080
ii ➥ 0.120
iii ➥ 0.200
iv ➥ 0.216

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Answer: III
Explanation:
Calculation,
The probability of a toothache, given evidence of a cavity, P(toothache | cavity) is ____.
P(cavity ) = 0.108 + 0.012 + 0.072 + 0.008 = 0.2

So, Option(III) is correct.

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Q12➡ | NET June 2021
Next five questions are based on the following passage.
Consider a domain consisting of three Boolean variables Toothache, Cavity, and Catch. The full joint distribution is a 2×2×2 table as shown in the figure below.

The probability of a cavity, given evidence of a toothache, P(cavity | toothache) is ____.
i ➥ 0.216
ii ➥ 0.280
iii ➥ 0.400
iv ➥ 0.600

Show Answer With Best Explanation

Answer: IV
Explanation:
Formula,
Conditional Probability
The probability of a toothache, given evidence of a cavity, P(toothache | cavity) is ____.
Calculation,
The probability of a cavity, given evidence of a toothache,
 The probability of a cavity, given evidence of a toothache, P(cavity | toothache) is ____.
 The probability of a cavity, given evidence of a toothache, P(cavity | toothache) is ____.
P(cavity Ո toothache ) = 0.108 + 0.012 = 0.12

P(cavity U toothache) is________________.
P(toothache) = 0.108 + 0.012 + 0.016 + 0.064 = 0.2
 The probability of a cavity, given evidence of a toothache, P(cavity | toothache) is ____.
 The probability of a cavity, given evidence of a toothache, P(cavity | toothache) is ____.= 0.6
So,Option(IV) is correct.

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Q13➡ | NET November 2020
Given below are two statements:
Statement I: 5 divides n5-n whenever n is a nonnegative integer.
Statement II: 6 divides n3-n whenever n is a nonnegative integer.

In the light of the above statements, choose the correct answer from the options given below
i➥ Both Statement I and Statement II are correct
ii ➥ Both Statement I and Statement II are incorrect
iii ➥ Statement I is correct but Statement II is incorrect
iv ➥ Statement I is incorrect but Statement II is correct

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Answer: I
Explanation:
Statement I: n5 – n
Example : Take n = 2 , 25- 2 = 32 – 2 = 30 , which is divided by 5.
Take n = 3 , 35 – 3 = 243 – 3 = 240 , which is divided by 5.
Statement I is true.
Statement II: n3 – n
Example : Take n = 2 , 23- 2 = 8 – 2 = 6 , which is divided by 6.
Take n = 3 , 33 – 3 = 27 – 3 = 24 , which is divided by 6.
Statement II is true.

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Q14➡ | NET November 2020
Find the lexicographic ordering of the bit strings given below based on the ordering 0 < 1.
A) 001
B) 010
C) 011
D) 0001
E) 0101
Choose the correct answer from the options given below:
i➥ 001<010<011<0001<0101
ii ➥ 0001<001<010<0101<011
iii ➥ 0001<0101<001<010<011
iv ➥ 001<010<0001<0101<011

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Answer: II
Explanation:
Lexicographically means sorting in natural order, dictionary order.
The lexicographic order of the given bit strings will be:
0001<001<010<0101<011
Take 0001 & 001 for comparing. Compare each bit of 1st string with the corresponding bit of 2nd string.

It is clear that 0001< 001.

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Q15➡ | NET November 2020
Consider the following statements:
A) Any tree is 2-colorable
B) A graph G has no cycles of even length if it is bipartite
C) A graph G is 2-colorable if is bipartite
D) A graph G can be colored with d+1 colors if d is the maximum degree of any vertex in the graph G
E) A graph G can be colored with O(log|v|) colors if it has O(|v|) edges.

Choose the correct answer from the options given below:
i➥ (C) and (E) are incorrect
ii ➥ (B) and (C) are incorrect
iii ➥ (B) and (E) are incorrect
iv ➥ (A) and (D) are incorrect

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Answer: III
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Q16➡ | NET November 2020
Let) G be a directed graph whose vertex set is the set of numbers from 1 to 100. There is an edge from a vertex i to a vertex j if and only if either j=i+1 or j=3i. The minimum number of edges in a path in G from vertex 1 to vertex 100 is ___________.
i➥ 23
ii ➥ 99
iii ➥ 4
iv ➥ 7

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Answer: IV
Explanation:
Edge set consists of edge from i to j , if and only if
either j=i+1
or j=3i
since, we to find the minimum number of edge from vertex 1 to vertex 100.so, we have to think about how we can reach an edge 100 from an edge 1 with minimum path.
1 ->3 ->9 -> 10 -> 11 -> 33 -> 99 -> 100.
We need minimum 7 edges.

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Q17➡ | NET November 2020
How many ways are there to pack six copies of the same book into four identical boxes, where a box can contain as many as six books?
i➥ 4
ii ➥ 6
iii ➥ 7
iv ➥ 9

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Answer: IV
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Q18➡ | NET November 2020
The number of positive integers not exceeding 100 that are either odd or the square of an integer is _.
i➥ 63
ii ➥ 59
iii ➥ 55
iv ➥ 50

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Answer: III
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Q19➡ | NET December 2019
Consider the following statements:
S1: If a group (G,) is of order n, and a ∈ G is such that am = e for some integer m ≤ n, then m must divide n.
S2: If a group (G,) is of even order , then there must be an element and a ∈ G is such that a ≠ e and a * a = e.

Which of the statements is (are) correct?
i➥ Only S1
ii ➥ Only S2
iii ➥ Both S1 and S2
iv ➥ Neither S1 nor S2

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Answer: III
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Q20➡ | NET December 2019
Let a2e mod n = (ae)2 mod n and a2c+1 mod n = a (ac)2 mod n. For a = 7, b = 17 and n = 561, what is the value of ab(mod n)?
i➥ 160
ii ➥ 166
iii ➥ 157
iv ➥ 67

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Answer: I
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Q21➡ | NET December 2019
Let P be the set of all people. Let R be a binary relation on P such that (a, b) is in R if a is a brother of b. Is R symmetric, transitive, an equivalence relation, a partial order relation?
i➥ NO.NO.NO.NO
ii ➥ NO.NO.YES.NO
iii ➥ NO.YES.NO.NO
iv ➥ NO.YES.YES.NO

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Answer: III
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Q22➡ | NET December 2019
What are the greatest lower bound (GLB) and the least upper bound (LUB) of the sets A = {3, 9, 12} and B = {1, 2, 4, 5, 10} if they exist in poset (z*,/)?
i➥ A (GLB – 3, LUB – 36); B (GLB – 1, LUB – 20)
ii ➥ A(GLB – 3, LUB – 12); B (GLB -1, LUB – 10)
iii ➥ A (GLB -1, LUB – 36); B (GLB – 2, LUB – 20)
iv ➥ A (GLB – 1, LUB – 12); B (GLB – 2, LUB – 10)

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Answer: I
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Q23➡ | NET December 2019
A clique in an undirected graph G = (V, E) is a subset V’ ⊆ V of vertices, such that
i➥ If (u, v) ∈ E then u ∈ V’ and v ∈ V’
ii ➥ If (u, v) ∈ E then u ∈ V’ or v ∈ V’
iii ➥ Each pair of vertices in V’ is connected by an edge
iv ➥ All pairs of vertices in V’ are not connected by an edge

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Answer: III
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Q24➡ | NET December 2019
How many reflexive relations are there on a set with 4 elements?
i➥ 24
ii ➥ 212
iii ➥ 42
iv ➥ 2

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Answer: II
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Q25➡ | NET June 2019
How many are there to place 8 indistinguishable balls into four distinguishable bins?
i➥ 70
ii ➥ 165
iii ➥ 8C4
iv ➥ 8P4 T

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Answer: II
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Q26➡ | NET June 2019
Suppose that a connected planar graph has six vertices, each of degrees four. Into how many regions is the plane divided by a planar representation of this graph?
i➥ 6
ii ➥ 8
iii ➥ 12
iv ➥ 10

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Answer: II
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Q27➡ | NET June 2019
The value of the derivative of the sigmoid function given by
f(x)= 1 / (1+e(-2x)) at x=0 is
i➥ 0
ii ➥ 1/2
iii ➥ 1/4
iv ➥

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Answer: II
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Q28➡ | NET June 2019
Consider the poset ({3,5,9,15,24,45},|).
Which of the following is correct for the given poset?
i➥ There exists a greatest element and a least element
ii ➥ There exists a greatest element but not a least element
iii ➥ There exists a least element but not a greatest element
iv ➥ There does not exist a greatest element and a least element

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Answer: IV
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Q29➡ | NET June 2019
Consider the following properties with respect to a flow network G=(V,E) in which a flow is a real-valued function f:VxV→ R
P1 : For all u,vεV, f(u,v)= -f(v,u)
P2 : ΣvεV f(u,v)=0 for all uεV

Which one of the following is/are correct?
i➥ Only P1
ii ➥ Only P2
iii ➥ Both P1 and P2
iv ➥ Neither P1 nor P2

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Answer: I
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Q30➡ | NET June 2019
A web application and its support environment has not been fully fortified against attack. Web engineers estimate that the likelihood of repelihood an attack is only 30 percent. The application does not contain sensitive or controversial information, so the threat probability is 25 percent. What is the integrity of the web application?
i➥ 0.625
ii ➥ 0.725
iii ➥ 0.775
iv ➥ 0.825

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Answer: IV
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Q31➡ | NET June 2019
For which values of m and n does the complete bipartite graph km,n have a Hamilton circuit?
i➥ m≠n, m, n≥2
ii ➥ m≠n, m, n≥3
iii ➥ m=n, m, n≥2
iv ➥ m=2, m, n≥3

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Answer: III
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Q32➡ | NET June 2019
How many cards must be selected from a standard deck of 52 cards to guarantee that at least three hearts are present among them?
i➥ 9
ii ➥ 13
iii ➥ 17
iv ➥ 42

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Answer: IV
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Q33➡ | NET June 2019
Find the zero-one matrix of the transitive closure of the relation given by matrix A:
Find the zero-one matrix ofthe transitive closure of the relation given by matrix A: A)	1 B)	2 C)	3 D)	4

1) Find the zero-one matrix ofthe transitive closure of the relation given by matrix A: A)	1 B)	2 C)	3 D)	4

2) Find the zero-one matrix ofthe transitive closure of the relation given by matrix A: A)	1 B)	2 C)	3 D)	4

3) Find the zero-one matrix ofthe transitive closure of the relation given by matrix A: A)	1 B)	2 C)	3 D)	4

4) Find the zero-one matrix ofthe transitive closure of the relation given by matrix A: A)	1 B)	2 C)	3 D)	4
i➥ 1
ii ➥ 2
iii ➥ 3
iv ➥ 4

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Answer: I
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Q34➡ | NET June 2019
Which of the following is principal conjunctive normal form for [(pVq) ∧ ~p → ~q] ?
i ➥ pV~q
ii ➥ pVq
iii ➥ ~p Vq
iv ➥ ~p V ~q

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Answer: I

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Q35➡ | NET June 2019
Match List-I with List-II
Match List-I with List-II List-I                          List-II (a) p → q                (i) ¬(q → ¬p) (b) p v q                  (ii) p ∧ ¬q (c) p ∧ q                  (iii) ¬p → q (d) ¬(p → q)            (iv) ¬p v q Choose the correct option from those given below: A)	(a)-(ii);(b)-(iii);(c)-(i);(d)-(iv) B)	(a)-(ii);(b)-(i);(c)-(iii);(d)-(iv) C)	(a)-(iv);(b)-(i);(c)-(iii);(d)-(ii) D)	(a)-(iv);(b)-(iii);(c)-(i);(d)-(ii)
Choose the correct option from those given below:
i ➥ (a)-(ii);(b)-(iii);(c)-(i);(d)-(iv)
ii ➥ (a)-(ii);(b)-(i);(c)-(iii);(d)-(iv)
iii ➥ (a)-(iv);(b)-(i);(c)-(iii);(d)-(ii)
iv ➥ (a)-(iv);(b)-(iii);(c)-(i);(d)-(ii)

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Answer: IV
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Q36➡ | NET December 2018
Consider the vocabulary with only four propositions A,B,C and D. How many models are there for the following sentence?
( ⌐ A ∨ ⌐ B ∨ ⌐ C ∨ ⌐ D)
i➥ 8
ii ➥ 7
iii ➥ 15
iv ➥ 16

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Answer: III
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Q37➡ | NET December 2018
The number of substrings that can be formed from string given by “a d e f b g h n m p” is
i➥ 10
ii ➥ 45
iii ➥ 56
iv ➥ 55

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Answer: III
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Q38➡ | NET December 2018
Consider the following statements related to AND-OR Search algorithm.
S 1 : A solution is a subtree that has a goal node at every leaf.
S 2 : OR nodes are analogous to the branching in a deterministic environment
S 3 : AND nodes are analogous to the branching in a non-deterministic environment.

Which one of the following is true referencing the above statements?
Choose the correct answer from the code given below:
i➥ S1- False, S2- True, S3- True
ii ➥ S1- True, S2- True, S3- True
iii ➥ S1- False, S2- True, S3- False
iv ➥ S1- True, S2- True, S3- False

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Answer: II
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Q39➡ | NET December 2018
The K-coloring of an undirected graph G=(V,E) is a function C: V➝{0,1,……,K-1} such that c(u)≠c(v) for every edge (u,v) ∈ E Which of the following is not correct?
i➥ G has no cycles of odd length
ii ➥ G has cycle of odd length
iii ➥ G is 2-colorable
iv ➥ G is bipartite

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Answer: II
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Q40➡ | NET December 2018
If a graph (G) has no loops or parallel edges and if the number of vertices(n) in the graph is n≥3, then the graph G is Hamiltonian if
(i) deg(v) ≥n/3 for each vertex v
(ii) deg(v) + deg(w) ≥ n whenever v and w are not connected by an edge.
(iii) E (G) ≥ 1/3 (n − 1 )(n − 2 ) + 2
i➥ (i) and (iii) only
ii ➥ (ii) and (iii) only
iii ➥ (iii) only
iv ➥ (ii) only

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Answer: IV
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Q41➡ | NET December 2018
A survey has been conducted on methods of commuter travel. Each respondent was asked to check Bus, Train or Automobile as a major methods of travelling to work. More than one answer was permitted. The results reported were as follows : Bus 30 people; Train 35 people; Automobile 100 people; Bus and Train 15 people; Bus and Automobile 15 people; Train and Automobile 20 people; and all the three methods 5 people. How many people complete the survey form ?
i➥ 160
ii ➥ 120
iii ➥ 165
iv ➥ 115

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Answer: II
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Q42➡ | NET December 2018
A full joint distribution for the Toothache, Cavity and Catch is given in the table below:
A full joint distribution for the Toothache, Cavity and Catch is given in the table below:  Which is the probability of Cavity, given evidence of Toothache ? A)	< 0.2, 0.8 srcset= B) < 0.6, 0.8 > C) < 0.4, 0.8 > D) < 0.6, 0.4 >“>
Which is the probability of Cavity, given evidence of Toothache ?
i➥ < 0.2, 0.8 >
ii ➥ < 0.6, 0.8 >
iii ➥ < 0.4, 0.8 >
iv ➥ < 0.6, 0.4 >

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Answer: IV
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Q43➡ | NET December 2018
In mathematical logic, which of the following are statements ?
(i) There will be snow in January
(ii) What is the time now ?
(iii) Today is Sunday
(iv) You must study Discrete Mathematics.
i➥ (iii) and (iv)
ii ➥ (i) and (ii)
iii ➥ (i) and (iii)
iv ➥ (ii) and (iv)

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Answer: III
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Q44➡ | NET December 2018
Consider the statements below :
“ There is a country that borders both India and Nepal. “
Which of the following represents the above sentence correctly ?
i➥ ∃c Border(Country(c), India ∧ Nepal)
ii ➥ ∃c Country(c) ∧ Border(c, India) ∧ Border(c, Nepal)
iii ➥ [∃c Country(c)] ⇒ [Border(c,India) ∧ Border(c, Nepal)]
iv ➥ ∃c Country(c) ⇒ [ Border(c, India) ∧ Border(c, Nepal)]

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Answer: II
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Q45➡ | NET December 2018
A box contains six red balls and four green balls. Four balls are selected at random from the box. What is the probability that two of the selected balls will be red and two will be green?
i➥ 1/35
ii ➥ 1/14
iii ➥ 1/9
iv ➥ 3/7

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Answer: IV
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Q46➡ | NET June 2018
In a multi-user operating system, 30 requests are made to use a particular resource per hour, on an average. The probability that no requests are made in 40 minutes, when arrival pattern is a poisson distribution, is
i➥ e -15
ii ➥ 1-e -15
iii ➥ 1-e -20
iv ➥ e -20

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Answer: IV
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Q47➡ | NET June 2018
Consider the following English sentence : “Agra and Gwalior are both in India”. A student has written a logical sentence for the above English sentence in First-Order Logic using predicate In(x, y), which means x is in y, as follows : In(Agra, India) ⋁ In(Gwalior, India) Which one of the following is correct with respect to the above logical sentence ?
i➥ It is syntactically valid but does not express the meaning of the English sentence.
ii ➥ It is syntactically valid and expresses the meaning of the English sentence also.
iii ➥ It is syntactically invalid but expresses the meaning of the English sentence.
iv ➥ It is syntactically invalid and does not express the meaning of the English sentence.

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Answer: I
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Q48➡ | NET June 2018
Consider the set of all possible five-card poker hands dealt fairly from a standard deck of fifty-two cards. How many atomic events are there in the joint probability distribution?
i➥ 2,598,960
ii ➥ 3,468,960
iii ➥ 3,958,590
iv ➥ 2,645,590

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Answer: I
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Q49➡ | NET June 2018
E is the number of edges in the graph and f is maximum flow in the graph. When the capacities are integers, the runtime of Ford-Fulkerson algorithm is bounded by:
i➥ O(E∗f)
ii ➥ O(E 2 ∗f)
iii ➥ O(E∗f 2 )
iv ➥ None of the above

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Answer: I
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Q50➡ | NET June 2018
Digital data received from a sensor can fill up 0 to 32 buffers. Let the sample space be S={0, 1, 2, ………., 32} where the sample j denote that j of the buffers are full and P (i) = (1/561)(33 − i ) . Let A denote the event that the even number of buffers are full. Then p(A) is:
i➥ 0.515
ii ➥ 0.785
iii ➥ 0.758
iv ➥ 0.485

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Answer: I
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Q51➡ | NET June 2018
The equivalence of ¬∃ x Q(x) is:
i➥ ∃x ¬Q(x)
ii ➥ ∀x ¬Q(x)
iii ➥ ¬∃ x ¬Q(x)
iv ➥ ∀x Q(x)

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Answer: II
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Q52➡ | NET June 2018
If A i = {−i, … −2,−1, 0, 1, 2, . . . . . i} then If A i = {−i, ... −2,−1, 0, 1, 2, . . . . . i} then is:  A	Z B	Q C	R D	C is:
i➥ Z
ii ➥ Q
iii ➥ R
iv ➥ C

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Answer: I
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Q53➡ | NET June 2018
If A i = {−i, … −2,−1, 0, 1, 2, . . . . . i} then If A i = {−i, ... −2,−1, 0, 1, 2, . . . . . i} then is:  A	Z B	Q C	R D	C is:
i➥ Z
ii ➥ Q
iii ➥ R
iv ➥ C

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Answer: I
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Q54➡ | NET June 2018
Which of the relations on {0, 1, 2, 3} is an equivalence relation?
i➥ { (0, 0) (0, 2) (2, 0) (2, 2) (2, 3) (3, 2) (3, 3) }
ii ➥ { (0, 0) (1, 1) (2, 2) (3, 3) }
iii ➥ { (0, 0) (0, 1) (0, 2) (1, 0) (1, 1) (1, 2) (2, 0) }
iv ➥ { (0, 0) (0, 2) (2, 3) (1, 1) (2, 2) }

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Answer: II
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Q55➡ | NET June 2018
Which of the following is an equivalence relation on the set of all functions from Z to Z?
i➥ { (f , g ) | f (x) − g (x) = 1 ∀ x ∈ Z }
ii ➥ { (f , g ) | f (0) = g (0) or f (1) = g (1)}
iii ➥ { (f , g ) | f (0) = g (1) and f (1) = g (0)}
iv ➥ { (f , g ) | f (x) − g (x) = k f or some k ∈ Z }

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Answer: IV
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Q56➡ | NET June 2018
Which of the following statements is true ?
i➥ (Z, ≤ ) is not totally ordered
ii ➥ The set inclusion relation ⊆ is a partial ordering on the power set of a set S
iii ➥ (Z, ≠ ) is a poset
iv ➥ The directed graph Which of the following statements is true ? A	(Z, ≤ ) is not totally ordered B	The set inclusion relation ⊆ is a partial ordering on the power set of a  set S C	(Z, ≠ ) is a poset D	The directed graph   is not a partial orderis not a partial order

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Answer: II
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Q57➡ | NET November 2017 Paper 2
If the time is now 4 O’clock, what will be the time after 101 hours from now?
i➥ 9 O’Clock
ii ➥ 8 O’Clock
iii ➥ 5 O’Clock
iv ➥ 4 O’Clock

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Answer: I
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Q58➡ | NET November 2017 Paper 2
Let A = Let A =   and B =   Find the boolean product A⊙B of the two matrices. and B =Let A =   and B =   Find the boolean product A⊙B of the two matrices.
Find the boolean product A⊙B of the two matrices.
i➥ Let A =   and B =   Find the boolean product A⊙B of the two matrices.
ii ➥ Let A =   and B =   Find the boolean product A⊙B of the two matrices.
iii ➥ Let A =   and B =   Find the boolean product A⊙B of the two matrices.
iv ➥ Let A =   and B =   Find the boolean product A⊙B of the two matrices.

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Answer: I
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Q59➡ | NET November 2017 Paper 2
How many distinguishable permutations of the letters in the word BANANA are there?
i➥ 720
ii ➥ 120
iii ➥ 60
iv ➥ 360

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Answer: III
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Q60➡ | NET November 2017 Paper 2
Let P and Q be two propositions, ¬ (P ↔ Q) is equivalent to:
i➥ P ↔ ¬ Q
ii ➥ ¬ P ↔ Q
iii ➥ ¬ P ↔ ¬ Q
iv ➥ Q → P

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Answer: I,II Both (Marks to all)
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Q61➡ | NET November 2017 Paper 2
Negation of the proposition ∃x H(x) is:
i➥ ∃x ⌐H(x)
ii ➥ ∀x ⌐H(x)
iii ➥ ∀x H(x)
iv ➥ ⌐x H(x)

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Answer: II
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Q62➡ | NET November 2017 Paper 3
Let P, Q, R and S be Propositions. Assume that the equivalences P ⇔ (Q ∨ ¬ Q) and Q ⇔ R hold. Then the truth value of the formula (P ∧ Q) ⇒ ((P ∧ R) ∨ S) is always:
i➥ True
ii ➥ False
iii ➥ Same as truth table of Q
iv ➥ Same as truth table of S

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Answer: I
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Q63➡ | NET November 2017 Paper 3
“If X, then Y unless Z” is represented by which of the following formulae in propositional logic?
i➥ (X ∧ Y) → ¬ Z
ii ➥ (X ∧ ¬ Z) → Y
iii ➥ X → (Y ∧ ¬ Z)
iv ➥ Y → (X ∧ ¬ Z)

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Answer: II
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