Download AKTU B-Tech 3rd Sem 2017-2018 RCS 301 Discrete Structures And Theory Of Logic Question Paper

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Paper ID: 1002

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Roll No.

Sub Code: RCS 301

B.Tech.

(SEM III) THEORY EXAMINATION 2017-18

Discrete Structures & Theory of Logic

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Total Marks: 70

Note: 1. Attempt all Sections. If require any missing data; then choose suitably.

2. Any special paper specific instruction.

SECTION A

1. Attempt all questions in brief. 2 x 7 = 14

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1. Define Eulerian path, circuit and graph
2. Let A=(2,4,5,7,8)=B,aRb if and only if a+b<=12.Find relation matrix
3. Explain edge coloring and k edge coloring.
4. Define Chromatic number and Isomorphic graph.
5. Define union and intersection of multiset and find for A=[1,1,4,2,2,3],B=[1,2,2,6,3,3].

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6. Find the contrapositive of –“If he has courage, then he will win”.
7. Define rings and write its properties.

SECTION B

2. Attempt any three of the following: 7 x 3 = 21

1. Prove by mathematical induction 3+3³+3³³+... = (10ⁿ⁺¹-9n-10)/27

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2. Define the following with one example:
   1. Bipartite graph.
   2. Complete graph.
   3. How many edges in K7 and K3,6
   4. Planar Graph.

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3. For any positive integer D36, then find whether (D36,'|') is lattice or not?
4. Let X={1,2,3.....7} and R={(x,y) | (x-y) is divisible by 3}.Is R equivalence relation. Draw the diagraph of R
5. Simplify the following Boolean function using K-map: F(x,y,z)=∑(0,2,3,7)

SECTION C

3. Attempt any one part of the following: 7x1=7

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1. (a) Solve a_r-6a_r-1+8a_r-2=r.4^r, given a₁=8, and a₀=1.
2. (b) Show that: r → ~q, r v s, s → ~q, p → q ⊢ ~p are inconsistent

4. Attempt any one part of the following: 7x1=7

1. (a) Write the properties of Group. Show that the set(1,2,3,4,5)is not group under addition and multiplication modulo 6.
2. (b) Prove by mathematical induction n⁴-4n² is divisible by 3 for all n>=2.

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5. Attempt any one part of the following: 7x1=7

1. (a)Explain Modular lattice, distribute lattice and bounded lattice with eg and diagram
2. (b) Draw the Hasse diagram of (A, ≤), where A= {3,4,12,24,48,72} and relation ≤ be such that a ≤ b if a divides b

6. Attempt any one part of the following: 7x1=7

1. (a) Given the inorder and postorder traversal of a tree T
   

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   Inorder: HFEABIGDC Postorder : BEHFACDGI
   Determine the tree T and its Preorder.
2. (b) Translate the following sentences in quantified expressions of predicate logic.
   1. All students need financial aid.
   2. Some cows are not white.
   3. Suresh will get first division if and only if he gets first div.

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   4. if water is hot,then shyam will swim in pool.
   5. All integer are either even or odd integer.

7. Attempt any one part of the following: 7x1=7

1. (a) Define and Explain any two the following:
   1. BFS and DFS in Trees.

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   2. Euler Graph
   3. Adjacency matrix of a graph.
2. (b) Solve the recurrence relation: a_r+4a_r-1 + 4a_r-2= r².

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