Download Sathyabama Univarsity B.E April 2010 Discrete Mathematics

SATHYABAMA UNIVERSITY
(Established under section 3 of UGC Act,1956)
Course & Branch :B.E - P-CSE
Title of the Paper :Discrete Mathematics
Sub. Code :611PT501
Date :22/04/2010
Max. Marks :80
Time : 3 Hours
Session :FN
______________________________________________________________________________________________________________________
PART - A
Answer ALL the Questions
(10 x 2 = 20)
1.
Find the principal disjunctive normal form of
(q  p)  (~ p  q).
2.
Define tautology and contradiction.
3.
Draw the Hasse Diagram for the relation “less than” on the set
A = {2, 3, 4, 6, 12, 36, 48}. Here a R b if a less than b for
a, b A.
4.
If A, B, C are sets such that A  B = A  C and A  B = A  C,
show that B = C.
5.
State Lagrange’s theorem.
6.
Find the product and verify the commutative law for the
1 2 3 4 
1 2 3 4 




f

andg

 4 3 1 2
permutations
3
1
4
2




7.
How many three digit numbers can be formed with the number 5,
7, 9, 1.
(a) A digit cannot appear more than once.
(b) Any digit may appear any number of times.
8.
Define a lattice with an example.
9.
Define a complete graph Kn. For what values of n, Kn is regular?
10. Draw a rooted tree for the given expression and also find the
value of the expression: + - * 2 3 5 /  234
PART – B
Answer All the Questions
(5 x 12 = 60)
11. (a) Using the method of direct proof, prove that,
r  ~ q, r  s, s  ~ q p  q  ~ p.
(b) Construct the truth table for the given propositions:
(p  q)  (~ p  r)
(or)
12. (a) Using the rules of predicate calculus, show that
(x) (P(x)  Q(x)), (x)P(x)  (x) Q(x).
(b) By using any method of proving, show that R  S follows
logically from the premises C  D, (C  D)  ~ H, ~ H  (A 
~ B) and (A  ~ B)  (R  S).
13. (a) Let X = {1, 2, 3,….,25} and R = {(x, y)/ x – y is divisible by
5} be a relation in X. Show that R is an equivalence relation.
(b) Let A, B, and C be three sets, f: A  B, g: B  C,, if f and g
are Surjective. Prove that gof is also surjective.
(or)
14. (a) Consider the relation R defined on the set {1, 2, 3, 4, 5, 6}and
R = {(i, j): | i – j| = 1}. Draw the graph representing the given
relation. Check whether it is reflexive, symmetric or transitive.
(b) Without using Venn diagram, prove A – (BC) = (A – B) 
(A – C)
15. (a) Prove that every subgroup of a cyclic group is cyclic.
(b) Prove that a group G is abelian iff (a * b)2 = a2 * b2.
(or)
16. (a) Prove that a subgroup H of a group G is normal iff x H x-1 = H
for all x  G.
(b) Prove that every cyclic group is abelian.
17. (a) Solve the recurrence relation
T(k) – 7T(k – 1) + 10T(k – 2)= 6 + 8k, k  2, with T(0) = 1
and T(1) = 2.
(b) Consider the Boolean function. f(x1, x2, x3) = [(x1  x2)  x3)]
 (x1  x2)
(i)
(ii)
(iii)
Simplify f algebraically.
Draw the switching circuit of f.
Draw the same of f obtained in (i).
(or)
18. (a) Using mathematical induction, prove that 1. 1! + 2.2! +
……..+ n.n! = (n + 1)! – 1, where ‘n’ is a positive integer.
(b) Using the recurrence relation, model the problem and solve.
Suppose that a person deposits Rs. 10,000 in a savings account at
a bank yielding 11% per year with interest compounded annually.
How much will be in the account after 30 years?
19. (a) If exists, find an example for the following. Justify.
(i) A weakly connected graph but not strongly connected.
(ii) A strongly connected graph but not weakly connected.
(iii) Both weakly connected and strongly connected.
(iv) Neither weakly nor strongly connected.
(b) How many edges are there in a forest of ‘t’ trees containing a
total of ‘n’ vertices?
(or)
20. (a) Prove that an undirected graph has even number of vertices
of odd degree.
(b) Prove that a graph G with ‘n’ vertices is a tree iff it has (n – 1)
edges.