Q. Given the following statements :

(i) Recursive enumerable sets are closed under complementation.
(ii) Recursive sets are closed under complementation.

Which is/are the correct statements ?

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Q. Let θ(x, y, z) be the statement “x + y = z” and let there be two quantifications given as

(i) ∀x ∀y  Z ∃ θ(x, y, z)
(ii) ∃Z ∀x ∀y θ(x, y, z)
 
Where x, y, z are real numbers. Then which one of the following is correct ?

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Q. Let P(rn, n) be the statement if "m divides n" where the universe of discourse for both the variables is the set of positive integers. Determine the truth values of each of the-following propositions:

I. ∀m ∀n P(m,n)
II. ∃m  ∀n P(m,n)

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Q. Big O estimate for
( f(x) = (x + 1) log(x2 + 1) +3x2 is given as

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Q. How many edges are there in a forest of t-trees containing a total of n vertices?

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Q. Let f and g be the functions from the set of integers to the set integers defined by

f(x) = 2x + 3 and g(x) = 3x + 2

Then the composition  of f and g and g and f is given as

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Q. A graph is non-planar if and only if it contains a subgraph homomorphic to

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Q. If the primal Linear Programming problem has unbounded  solution, then it's dual problem will  have

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Q. Given the problem to maximize
            f(x), X =(x1, x2 ,..... .xn)
subject to m number of inequality constraints
  gi(x) ≤ bi  , i = 1, 2 ...... m
including the non-negativity constraints x ≥ 0
Which of the following conditions is a Kuhn-Tucker necessary condition for a local maxima at x  ?

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Q. The following Linear Programming problem has :
Max                                   Z =x1 +x2
Subject to                         = x1-x2 ≥ 0                                  
                                             3x1 -x2 ≤ -3                                
      and                              xI , x2  ≥ 0

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Q. Given a flow graph with 10 nodes,13 edges and one connected components, the number of regions and the number  of predicate (decision) nodes in the flow graph will be

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Q. If h* represents an estimate of the cost of getting from the current node N to the goal node and h represents actual cost of getting from the current node to the goal node, then A*algorithm gives an optimal solution if

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Q. The recurrence relation T(n) = m T(n/2) tan2 is satisfied by

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Q. A ___________ complete subgraph and a __________ subset of vertices of a graph G = (V, E) are a clique and a vertex cover respectively.

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Q. A certain tree has two vertices of degree 4, one vertex of degree 3 and one vertex of degree 2. If the other vertices have degree 1, how many vertices are there in the graph?

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Q. Consider a set A {1, 2, 3,……… 1000}. How many members of A shall be divisible by 3 or by 5 or by both 3 and 5?

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Q. A computer program selects an integer in the set {k : 1 < k < 10,00,000} at random and prints out the result. This process is repeated 1 million times. What is the probability that the value k = 1 appears in the printout at least once?

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Q. Which one of the following is used to compute cyclomatic complexity?

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Q. If we define the functions f, g and h that map R into R by :

f(x) = x4, g(x) = √x2 + 1, h(x) = x2 + 72,  then the value of the composite functions ho(gof) and (hog)of are given as

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Q. Dijkstra algorithm, which solves the single-source shortest-paths problem, is a_______________, and the Floyd-Warshall algorithm, which finds shortest paths between all pairs of vertices, is a _____________.

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