Hindustan Institute of Technology and Science (HITE) 2009 B.E Computer Science University - Question Paper

DiscRETE MATHEMATICS
PART A:

1.Prove that the only idempondent elements of a group is its identity elements?
2.If A has three elements and B has two elements,then how many functions f:A->B can there be?
3.Let the function f:R->R be described by f(x)=x3 -2.Find f inverse?
4.Define Recursion function and primitive Recursion function?
5.Define ring.Give an example?
6.Find all non-trivial subgroups of(z6,+6)?
7.Define a field?
8.Define a Boolean algebra?
9.If f=(1234 and g=(1234 are permutations,prove that (gof) inverse=f inverse o g inverse?
3214) 2341)
10.Define invertible functions,characterictics functions?

PART B

11.(a)Explain classification and kinds of functions with example?
(b)Prove that the composite of 2 bijection is also bijection?
(c)Show that A interscetion(B union C)=(A intersection B)union (A intersection C)?

OR
(d)Show that f(x,y)=x+y is primitive recursive and Hence calculate the value of f(2,4)?
(e)Show that f(x,y)=xy is primitive recursive function?

12.(a)State and prove Lagranges theorem?
(b)Let f(G,*)->(H,Delta) be group homomorphism.Then show that Ker(f) is a normal group?
OR
(c)Obtain all the elements of(s3,poset) and also construct the composition table with respect to the operation of poset and show that (s3,poset)is not an abelian group?
13.(a)Show that every totalyy odered set is Lattice?
(b)Let R denote a relation on the set of all ordered pairs of positive integers by(x,y)R(u,v)iff xv=yu?
OR
(c)Show that if L is distributive Lattice then for all a,b,c belongs to L?
(a*b)+(b*c)+(c*a)=(a+b)*(b+c)*(c+a)
(d)If (L,*,+) is a distributive lattice and if a*b=a*c and a+b=a+c for all a,b,c belongs to L.Show that b=c and hence show that complement of an element is unique if it exist in L?
14.(a)If f is homomorphism of a group G into a Group G complement,then,(1)Group homomorphism preserves identity,(2)group homomorphism preserves inverse?
(b)State and prove fundamental theroem on homorphism of group?
OR
(c)A subgroup H of G is normal iff every left coset of H in G is equal to the right coset of H in G?
(d)The intersection of any 2 normal subgroup of a group is a normal subgroup?
15.(a)Establish demorgan's laws in a boolean algebra?
(b)Let(L,<=)be a lattice.For any a,b belongs to L,a<=b<->a intersection b=a<->a union b=b?
OR
(c)If f:A->B and g:B->C are bijection,prove that (gof)inverse=f inverse o g inverse?
(d)Show that f(x+y)=x+y,x,y belongs N is primitive recursive?

Earning:   Approval pending.