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Cusat Cs 3rs Sem

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• 8/6/2019 Cusat Cs 3rs Sem

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BTS(C) - III - 08 - 064 - BB. Tech Degree III Semester Examination, November 2008

IT/CS 303 DISCRETE COMPUTATIONALSTRUCTURES(2006 Scheme)

Time: 3 Hours Maximum Marks : 100PART A

(Answer ALL questions)(Each question carries FIVE marks)

I. (a)(8 x 5 = 40)

Using Mathematical Induction, verify that the equation is true for every positive integer n.1 1 1 1 n1 . 3 + 3 . 5 + 5.7 + . .. .. .+ (2n-I)(2n+ 1) =2n + 1 .

(b) Let fbe a function from X ={O , I, 2 , 3 , 4} to X defined byf (x}=4x mod 5

Writejas a set of ordered pairs and draw the arrow diagram off Isjone to one or onto?(c) Explain recursive algorithms with an example.

. II(d) Use the Binomial Theorem to show that L2 k c( n, k ) = 3 " .

k =O(e) Explain Hamiltonian cycles. Show that the graph given below does not contain a

Hamiltonian cycle.

(f) Decide whether the graph has an Euler cycle. If the graph has one, exhibit one.b fl, f

c. . C j ~ "k(g) Consider the set A = {k, I, m , n , p } and the corresponding relation.R = { ( k , k ) ,(/ ,/ ),( m.m ).( n,n )(p ,p ).( k .m),( k./ ) ,

(k , n ),( k ,p ),( m.n ),( m, p ),(n,p ),(l,p )}Construct the directed graph and corresponding Hasse diagram of this partial order.

(h) Let (A , *) be a semigroup. For every a and b in A, if a;t: b then a * b *- b * a anda=a= a .

(a) Show that for every a,b in A a * b * a = a(b) Show that for every a, b, c in A a * b * c :::: * c .

(Tum Over)

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II.

(e)

Ill. (a)(b)

IV. (a)(b)

V . (a)

VI.

(b)(c)(a)

(b)

2 ", IPARTB

(a)(b)

(4 x 15'" 60)--.---If p , q and r are true, find the truth value of the proposition (p V q) /\ P /\ r V q . (3)Determine whether the relation defmed on the set of positive integers is reflexive,symmetric, antisymmetric, transitive and/or a partial order

(i) (x,y) E R if 2 divides x+ y(ii) (x,y) E R if3 divides X+ y. (4)

Let R be the relation onthe set of eight - bit strings by s.Rs, provided that 51 and S2have the same number of zeros.

(i) Show that R is an equivalence relation(ii) How many equivalence classes are there?(iii) List one member of each equivalence class.

OR(8)

State and prove the generalized De Morgan's laws for logic.Consider A = B = C = R and let f :A -)0 Band g :B -)0 C be defined byf ( ) = X +9 and g( y ) = y2 + 3. Find the following composition functions.(i) fof(a) (ii) gog (a) (iii) fog (b)(iv) gof(b) (v) gof( 4 ) (vi) fog (-4). (9)

(6)

ORDifferentiate between analysis and complexity of algorithms. Explain the differentnotations for representing complexity.Find the complexity of 2n +3 log, n .

ORSolve the recurrence relation

an = -8an_1 -16an_2subject to initial conditions ao =2, a 1 =-20.State the Pigeon Hole Principle.Two fair dice are rolled. Find the probability of getting doubles or the sum of 5.

(10)(5)

(7)(5)(3)

Find the length of the shortest path and a shortest path between each pair of verticesin the weighted graph. b :G

h Q.D" I(i) a,f (, (ii) bj JDraw the undirected graph represented by incidence matrix Ml as shown below.

(10)

el e2 e3 e4 es e6a 0 1 0 0 1 1b 0 1 1 0 1 0c 0 0 0 1 0 1d 1 0 0 0 0 0

(5)e 1 0 0 1 0 0

OR(Contd .. .3)

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VII. (a)

(b)

VIII. (a)

IX.

3Find a minimal spanning tree using Prism's algorithm.0 - - - 3. b -4 c . . (10)

1 0 :~ 4 -4- .,~,~3

Find a spanninttreefor ~e grap~. !Z . . Q , _0....> . 1 : 1 " . c ,_ ' . ' ; ' " q ( .o~ _ . l X 1 > < I /. ~ k' .tL. . 1 . . . .Determine whether th~Dosets shown below are lattices or not.~ r 1M

(5). . , ..,(i) (ii) b c

2 3

(iii) (10)< 1 >; ( . ' . :..aL .. ' b ~iv)(b)

-\,Consider the set A ={1,2,5,lO,25,50} and the relation divides ( I ) be a partialordering relation on A.

(i) Draw the Hasse diagram of A with relation divides.(ii) Determine all upper bounds of 5 & 10(iii) Determine all lower bounds of 5 & 10.

OR(5)

Consider an algebraic system (Q , * ) where Q is the set of rational numbers and *is a binary operation defmed by a * b =a +b - aba, b EQ. Determine whetherI(Q , * ) is a group. . (7)(b) Let A = {a,b } . Which of the following tables define a semigroup on A? Which

(a)

of them defme a monoid on A?(i) * a ba b b. b a a(ii) * a ba a bb b b

1 1

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