If A = x x = 3 n , n 6 , n N B = x x = 9 n , n 4 , n N then find A B , A Band A − B03 .
[ marks]Check whether the relation Rdefined in the 1 , 2 , 3 , 4 , 5 , 6 as R= (a,b):b=a+1 is reflexive, symmetric or transitive.
[4 marks](I) Prove that p → q and ( p q ) are logically equivalent. (II) Let Y = n 2 : n N and f : N → Ydefined as f(n)=n2. Show that f is invertible and find the inverse of f04 .
[3 marks]In a survey of 400 students in a school, 100 were found as drinking apple juice, 150 as drinking orange juice and 75 drinking both apple and orange juice. Find how many students drink neither apple nor orange juice.
[3 marks]Let f :R→Rbe function defined by f ( x ) = a x + b . Find values of a and b04 for which f f =I R
[ marks]If R = ( a , b ) : a − b = 1 and S = ( a , b ) : a − b i s e v e n be two relations on A={1,2,3,4}. Then (I) Find matrices of Rand S , (II) Find digraph of Rand S (III) Find the relation R S07
[ marks](I) Find n if nC + n+2C = nP333 (II) From 4 professors and 6 students, a committee of 3 is to be formed. In how many ways, this can be done, if the committee contains (1) at most 1 professor (2) at least 2 professors.
[4 marks]Find reachable set of all the vertices and node base of following graph.
[3 marks]Show that in a lattice if ab c, then (I)2 a b = b c and (II) ( a b ) ( b c ) = b = ( a b ) ( a c )04
[ marks]Solve the recurrence relation which represents the Fibonacci sequence F n = F n − 1 + F n − with F0 = F1 = 107 .
[2 marks]The Indian cricket team consist of 16 players. It includes 2 wicket keepers and 5 bowlers. In how many ways eleven player can be selected if we have to select on wicket keeper and at-least 4 bowlers?
[3 marks]Prove that any graph has even number of odd vertices.
[4 marks]Find the generating function of a =a +a , where a =a =1 for n+2 n+1 n 0 1 n 107
[ marks]If Gis an abelian group with n elements g ,g ,.....,g then show that 1 2 n ( g1 g2 . . . . . g n ) 2 = e , where e03 is the identity element of G
[ marks]Draw a binary tree whose post-order produced the string d − g − e − b − i − j − h − f − c − a and pre-order produces the string j − b − a − c − d − i − h − e − g − f04
[ marks]Define Lattice. And draw the Hasse diagram representing the partial ordering ( A , B ) : A B on the power set P ( S ) where S = { 1 , 2 , 3 }07 . Find the maximal, minimal, greatest and least elements of the poset.
[ marks]Define Isolated vertex, Pendent vertex and Size of a graph
[3 marks]LetG be a graph with n vertices and m edges such that vertices have degree k or k + 1 . Prove that ifG has vertices of degree and N k + 1 vertices of degree k +1then N k = ( k + 1 ) n − 2 m04
[ marks]Show that Gis an abelian group under usual matrix addition, where a b G= a,b,c,d R c d
[7 marks]Show that the only right coset of a subgroup Hin a group Gthat is also subgroup of G03 is Hitself.
[ marks]Dose there exists a 4- regular graph with 6 vertices? If so, construct the graphs.
[4 marks]Show that ( R , + , ) is an integral domain, whereR= a+b 5 a,bI
[7 marks]Define cycle, walk and tree.
[3 marks]Find transitive closure by Warshall’s Algorithm ifA={1,2,3,4,5} and R = { ( 1 , 2 ) , ( 3 , 4 ) , ( 4 , 5 ) , ( 4 ,1 ) , ( 1 ,1 ) }04
[ marks]Define Isomorphic Graphs. Determine whether the following graphs are isomorphic or not. N k k3
[7 marks]