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)
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)]
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?
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
Show Answer With Best Explanation
Answer: IV Explanation: Formula, Conditional Probability Calculation, The probability of a toothache, given evidence of a cavity, P(toothache Ո cavity ) = 0.108 + 0.012 = 0.12
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
Show Answer With Best Explanation
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) P(cavity ) = 0.108 + 0.012 + 0.072 + 0.008 = 0.2
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
Show Answer With Best Explanation
Answer: I Explanation: Formula, Calculation, P(cavity Ո(toothache U catch)) = 0.108 + 0.012 + 0.072 = 0.192
` 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 _____.
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 Calculation, The probability of a cavity, given evidence of a toothache, P(cavityՈtoothache ) = 0.108 + 0.012 = 0.12
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
Show Answer With Best Explanation
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.
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
Show Answer With Best Explanation
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.
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:
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
Show Answer With Best Explanation
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.
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?
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.
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?
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*,/)?
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?
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
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?
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)
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:
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?
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
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 ?
Q42➡|NET December 2018 A full joint distribution for the Toothache, Cavity and Catch is given in the table below: 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 ?
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.
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)]
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?
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
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.
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?
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:
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:
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:
= 0.4615
= 0.6
B) < 0.6, 0.8 > C) < 0.4, 0.8 > D) < 0.6, 0.4 >“>
is:
is not a partial order
and B =
