Download AKTU B-Tech 8th Sem 2016-17 Discrete Mathematics Question Paper

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B. TECH.

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THEORY EXAMINATION (SEM–VIII) 2016-17

DISCRETE MATHEMATICS

Time: 3 Hours Max. Marks : 100

Note: Be precise in your answer. In case of numerical problem assume data wherever not provided.

SECTION – A

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1. Attempt all parts of the following question: 10 x 2 = 20

   1. What is the difference in relation and function?

   2. Define equivalence relation.

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   3. Transform following statement into symbolic form: Jack and Jill went up the hill.

   4. Define negation.

   5. Find the permutations of the set A = {1,2,3,4} taking two at a time.

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   6. There are 10 different people at a party. How many ways are there to pair them up into a collection of 5 parings ?

   7. Show that (I,+) is an abelian group.

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   8. Define cyclic group.

   9. Define Hamiltonian Path.

   10. Define Chromatic number.

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SECTION – B

1. Attempt any five parts of the following questions: 5 x 10 = 50

   1. Let R be the relation on the set A of integers, defined by xRy if x - y is divisible by 4. Show that R is an equivalence relation, and describe the equivalence classes.

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   2. Show the implication

      1. (P?¬P)?Q?(Pv¬P)?R
      2. (P?Q)?Q?PvQ

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   3. Find the solution of recurrence relation a_n = 6a_n-1+11a_n-2-6a_n-3 with condition a₀ = 2,a₁ = 5 and a₂ =15

   4. Show that (F,+,.) is a field where F is a set of all rational numbers and + and . are ordinary addition and multiplication operators.

   5. Show that number of odd degree vertices is always even.

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   6. Show that the graph shown in figure does not contain Hamiltonian Circuit.

      [Diagram description: A graph with vertices a, b, c, d, e, f, with edges connecting a-b, b-c, c-d, d-e, e-f, f-a, a-e, b-d, c-f.]

   7. Let G be a group; for fixed element x ? G, let G_x = {a?G:ax = xa} show that G_x is a subgroup of G for all x ? G.

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   8. Determine the generating function of the numeric function a_r where (i) a_r = 3^r +4^r+1, r=0 (ii) a_r =5, r=0

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SECTION - C

1. Attempt any two parts of the following questions: 2 x 15 = 30

   1. 1. Show that A?(BnC)=(A?B)n(A?C) Using Venn Diagram.

      2. Show that whether the relation (x, y) ? R, if x= y defined on the set of positive integer is partial order relation.

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   2. 1. Consider an algebraic system (G,*) where G is the set of all non-zero real numbers and * is a binary operation defined by a*b= ab/4 show that (G,*) is an abelian group.

      2. Prove that if H₁ and H₂ are two subgroups of G, then H₁nH₂ is also a subgroup.

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   3. 1. State and prove Hand Shaking Lemma.

      2. Show that maximum number of edges in a simple graph with n vertices is n(n-1)/2

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