Paper - IMCA 4 : DISCRETE MATHEMATICAL ... - Surana college

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I Semester M.C.A. Degree Examination, February/lVlarch 20ll
(Y2K5 Scheme)
IMCA 4 : DISCRETE MATHEMATICAL STRUCTURES
Time : 3 Hours
Instruction : Answer the questions in Parts as per the instructions.
PART - A
Max. Marks : g0
1. Answer any ten questions. Each question carries two marks. (2x10=20)
a) Define a power set. Illustrate with an example.
b) Find the truth value of proposition q -+- p
where p: Jris an irrational number
q : All square are rectangle.
c) In the f(x) = x+l frorn set of integer to the set of integer onto ?
d) If f :R-+R and g:R+R aredefinedbyf(x) =2x2+l andg(x) =2x*1,
then find fog and gof.
e) Prove by mathematical induction.
1.2 + 2.3 +..
_ n(n + 1Xn + 2)
3
0 Define Fibonacci number
5
g) What is the value of t j' ?
F
h) In how many ways can be letters in the woTd "MISSISSIPPI" be arranged ?
i) Explain pigeon hole principle.
j) Define complete graph. Give an example.
k) Give two examples of a graph which is Lamiltonian but not Eulerian.
P.T.O.

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1)
m)
n)
If 'a' is a generator of a cyclic group then prove that a-l is also a generator.
Define automorphism for grouP.
Let G = {1,- 1, i, - i}be a group under multiplication. Find the order of an
element in a graph G.
o) Define maximal and minimal element of a partially ordered set (A,

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PG - 483
4. a) Let G be a connected digraph, then show that G is a Eulerian graph iff G is
balanced.
5
b)
c)
Define Hamiltonian graph. Give two examples.
Prove that there is one and only one path between every pair of nodes in the
tree.
4
PART _ C
Answer any two full questions.
(2x15=30)
5. a) Define subgroup. Show that the intersection of two subgroups of a group
G is a subgroup of G. 5
b) State and prove Langrange's theorem. 4
c) Define Cyclic group. Prove that every cyclic group is an abelian group. 6
6. a) Prove that a non-empty subset H of a group is a subgroup if and only if for
all a. 56Hi ab-le H. 5
b) Prove that, if 'a' is a genarator of a cyclic group, then so is a-r. 4
c) Define normal subgroup of G. Prove that the subgroup H of G is a normal
subgroup of G if and only if every left coset of H in G is a right coset of H in G. 6
7 . a) Let (A, v, ^)
be an algebraic system defined try a lattice (A, < ). Prove that
both the join and meet operations are commutative. 5
b) For any a,b, c, dinalattice (A,