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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