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Code No: NR/RR210501 II B.Tech. I-Semester Regular Examinations, November-2003 DISCRETE STRUCTURES AND GRAPH THEORY (Computer Science and Engineering, Computer Science and Information Technology, Computer Science and Systems Engineering, Electronics and Computer Engineering) Time: 3 hours Max. Marks: 80 Answer any FIVE questions All questions carry equal marks ---
Write the following statement in symbolic form: “The crop will be destroyed if there is a flood”.
b)
Construct the truth table of the following formula. ┐(PV(Q R))⇄ ((PVQ) (PVR).
2. a)
Let A be a set with cardinality n and let R be a relation on A. Then prove that the transitive closure R is given by R = R U R2 U ------------------ U Rn Let A = {1,2,3,4,5} and R = { (1,1), (1,2), (2,1), (2.2), (3,3), (3,4), (4,3), (4,4), (5,5) } and S = { (1,1), (2,2), (3,3), (4,4), (5,4), (4,5), (5,5) }. Find the smallestequivalence relation containing R and S and compute the partition of A that itproduces
3. a) b)
4. a)
Let dm and dM demote the minimum and maximum degrees of all the vertices of G(V,E), respectively. Show that, for a non directed graph G, dm 2. | E | / | v | dM Suppose that G is a non directed graph with 12 edges. Suppose that G has 6 vertices of degree 3 and the rest have degrees less than 3. Determine the minimum number of vertices, G can have.
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b)
If f : XY and g : YX the function g is equal to f-1 only if g f = Ix and f g = Iy .Prove the result . Let f : R R and g : R R where R is the set of real numbers. Find f g and g f where f(x) = x2 – 2, g(x) = x + 4. State whether these functions are injective, subjective or objective.
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b)
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1. a)
Km, n represents a complete be partite graph. a) Is there a Hamiltonian circuit in K4,6? b) Is there a Hamiltonian path in K4,5? c) State a necessary and sufficient condition on the existence of Hamiltonian circuit in Km,n .
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Code No. NR/RR-210501
7. a) b)
8.
What is a binary tree? Describe the applications of binary trees. Also describe necessary conditions to represent a binary tree. Represent a binary tree show by means of an example that a simple digraph in which exactly one node has in degree 0 and every other node has in degree 1 is not necessarily a directed tree.
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Explain the terms: i) Disjunctive counting and ii) Sequential counting. How many numbers can be formed using the digits 1,3,4,5,6,8, and 9 if no repetitions are allowed ? Solve the recurrence relation: T(k) + 3 kT(k-1) = 0, T(0) = 1.
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6. a)
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Set No.
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Code No: NR/RR210501
2. a)
b)
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II B.Tech. I-Semester Regular Examinations, November-2003 DISCRETE STRUCTURES AND GRAPH THEORY (Computer Science and Engineering, Computer Science and Information Technology, Computer Science and Systems Engineering, Electronics and Computer Engineering) Time: 3 hours Max. Marks: 80 Answer any FIVE questions All questions carry equal marks --1. Using the statements R: Mark is rich H: Mark is happy Write the following statement in symbolic form: a) Mark is poor but happy b) Mark is rich (or) unhappy. c) Mark is neither rich nor happy d) Mark is poor (or) he is both rich and unhappy.
Define a poset with an example. Also define glb and lub in a poset. If A = {a,b,c} such that (A) is a poset with a partial order on M and B = { {a,b},{a,c} } find the glb and lub of B. Let L be a poset under partial ordering . Let a,bL, then show that: i) if a and b have a lub, then this lub is unique. ii) if a and b have a glb, then this glb is unique. Show that the function f(x) = x3 and g(x) = x1/3 for xR are inverse of each other. Show that { < x, x > / x N } which defines the relation of equality is primitive recursive.
4. a)
Let G be a simple graph, all of whose vertices have degree 3 and | E | = 2 | V | - 3. What can be said about G? Can a simple graph with 7 vertices be isomorphic to its complement?
b)
Show that in a connected planar linear graph with 6 vertices and 12 edges, each of the regions is bounded by 3 edges.
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6. a)
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How many different directed trees are there with three nodes? How many different ordered trees are there with three nodes? Show that in a complete binary tree, the total no. of edges is given by 2(n-1), where nt is the number of terminal nodes.
7. a)
b)
8.
In how many ways can 10 people be seated in a row so that a certain pair of them are not next to each other ? Define the combinations and permutations. Solve the recurrence relation an-9an-1+26an-2-24an-3=0 for n≥3. ###
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3. a) b) 4. a) b) 5. a) b) 6. a)
If f : XY and g : YZ and both f and g are onto; show that g f is also onto. Is g f is one to one if both g and f are one to one? Justify. Let D(x) denote the number of divisions of x. Show that D(x) is primitive recursive. Show that the sum of the in - degrees over all vertices is equal to the sum of the out – degrees over all vertices in any directed graph. Let G = (V,E) be an undirected graph with K components and | V | = n, | E | = m. Prove that m n – k. What is the chromatic number of a i) Tree ii) Complete graph (Kn) iii) Cycle (Cn) State and explain various applications of chromatic numbers. From the adjacency matrix of a simple digraph, how will you determine whether it is a directed tree? If it is a directed tree, how will you determine its root and terminal nodes? What is “tree traversal”? What are the different tree traversal methods? Explain them in brief with suitable examples.
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b)
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Prove that the relation “congruence modulo m “ given by = { / x-y is divisible by m } over the set of positive integers is an equivalence relation. Let A be given finite set and (A) its power set. Let be the inclusion relation on the elements of (A). Draw Hasse diagram of < (A) . > for i) A = {a} ii) A = {a,b} iii) A = {a,b,c,d}
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2. a)
In how many ways can a hand of 5 cards be selected from a deck of 52 cards ? Suppose there are 15 red balls and 5 white balls. Assume that the balls are distinguishable and that a sample of 5 balls is to be selected. i) How many samples of 5 balls are there? ii) How many samples contain all red balls? iii) How many samples contain 3 red balls and 2 white balls? iv) How many samples contain at least 4 red balls?
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7. a) b)
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3
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Code No: NR/RR210501 II B.Tech. I-Semester Regular Examinations, November-2003 DISCRETE STRUCTURES AND GRAPH THEORY (Computer Science and Engineering, Computer Science and Information Technology, Computer Science and Systems Engineering, Electronics and Computer Engineering) Time: 3 hours Max. Marks: 80 Answer any FIVE questions All questions carry equal marks --1. Construct the truth tables for the following formula; (P Q) (┐P Q) (P ┐Q) (┐P ┐Q)
Solve the recurrence relation an-8an-1+21an-2-18an-3=0 for n≥3. ###
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Code No: NR/RR210501 II B.Tech. I-Semester Regular Examinations, November-2003 DISCRETE STRUCTURES AND GRAPH THEORY (Computer Science and Engineering, Computer Science and Information Technology, Computer Science and Systems Engineering, Electronics and Computer Engineering) Time: 3 hours Max. Marks: 80 Answer any FIVE questions All questions carry equal marks --1. a) Explain the well-formed formulae. b) Explain the terms. i) Tautology ii) Contradiction iii) Substitution instance.
Show that in a poset of real numbers with usual order , no element covers any other. b) Define the relation R on a set A of positive integers by (a, b) R if and only if a/b can be expressed in the form 2m where m is an arbitrary integer. Show that R is an equivalence relation. Also determine the equivalence classes under R.
3. a) b)
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2. a)
Show that for any fixed k the relation given by { | y > k } is primitive recursive. Show that the sets of even numbers and odd numbers are both recursive Define the terms ‘path’ and ‘transitive closure’. Prove that a digraph G is strongly connected iff there is a closed directed path containing every vertex in G.
5.
Prove that if a connected graph has 2n vertices of odd degree, then a) n Euler paths are required to contain each edge exactly once. b) There exists a set of n such paths.
6. a) b)
Write the Algorithm for traversing a tree in preorder. Give an example. What are the application areas where preorder traversal can be implemented? Give at least four examples.
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4. a) b)
In how many ways can 30 distinguishable books be distributed among 3 people A,B and C so that. i) A and B together receive exactly twice as many books as “C””? ii) C receives atleast 2 books, B receives at least twice as many books as C, and A receives at least 3 times as many books as B? How many different outcomes are possible by tossing 10 similar coins?
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7. a)
b)
8.
Explain the Recurrence relation. What is its application in computer science and with suitable examples? ###
