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MANIPAL INSTITUTE OF TECHNOLOGY (Constituent Institute of Manipal University) Manipal - 576 104 THIRD SEMESTER B.E DEGREE MAKE-UP EXAMINATION – JULY, 2009

SUB: DISCRETE MATHEMATICS (MAT – CSE – 201) (REVISED CREDIT SYSTEM)

Time : 3 Hrs.

Max.Marks : 50

 Note : a) Answer any FIVE full questions. b) All questions carry equal marks.

1A.

State the principle of inclusion and exclusion. Hence show that the proportion of permutations of 1,2,3,…..,n which contain no consecutive pair i, i + 1, for any i is approximately

n 1 . ne

1B.

For any graph G with 6 vertices, show that either G or G contains triangle.

1C.

Show that the number of partitions of n in which every part is odd is equal to number of partitions of n with unequal parts (4+3+3)

2A.

Given n = 5 and the five marks 0, 1, 2, 3, 4 what are the 50th and 100th permutations in the following orders (the first permutation is 01234) a) Reverse Lexicographical b) Fike’s ordering.

2B.

Show that in any graph G, number of vertices of odd degree is even

-1-

2C.

Show that P

Q,Q

R

P

is a valid conclusion from the premises

Q

M , and ‫ך‬M.

R,P

(4+3+3) 3A.

State and prove Lagrange’s theorem

3B.

Let E x1, x 2 , x3

x1 x3 be a Boolean expression over the two

x1 x 2

valued Boolean algebra. Write E x1, x 2 , x 3

in both disjunctive and

conjunctive normal forms. 3C.

Show that a (p, q) graph G is a tree iff it is connected and q = p – 1. (4+3+3)

4A.

Let A, , , A and b = a1

4B.

4C.

be a finite Boolean algebra. Let b be any non zero element in

a1, a2,…….,ak be all the atoms of A such that ai a2

…….

b. Prove that

ak .

Symbolise the following using predicates. (i)

For every positive integer there exists a greater positive integer.

(ii)

All elephants are grey

(iii)

x is the father of the mother of y.

Let (A, ) be a distributive lattice. If a a

x

a

y & a

x

a

y for all

A show that x = y.

(4+3+3) 5A.

Show that from (i)

( x) (F(x)

(ii)

( y) (M(y)

S(x))

(y) (M(y)

W(y))

‫ ך‬W(y)).

the conclusion (x) (F(x)

‫ך‬S(x)) follows.

-2-

a n b a m | m, n

5B.

Construct a grammar which generates the language L

5C.

Show that every group of order less than or equal to 5 is abelian.

1 .

(4+3+3) 6A.

Implement Dijkstraa’s algorithm to find shortest path from ‘c’ to all other vertices , for the following network.

6B.

Show that subgroup of a cyclic group is cyclic.

6C.

If the meet operation is distributive over the join operation in a lattice, then show that the join operation is also distributive over the meet operation.

(4+3+3) *****

-3-

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MANIPAL INSTITUTE OF TECHNOLOGY (Constituent Institute of MAHE – Deemed University) Manipal - 576 104 THIRD SEMESTER B.E DEGREE MAKE-UP EXAMINATION – 2005-06

SUB: DISCRETE MATHEMATICS(MAT – CSE – 201) (REVISED CREDIT SYSTEM)

Time : 3 Hrs.

Max.Marks : 50

 Note : a) Answer any FIVE full questions. b) All questions carry equal marks.

1A.

State the principle

of inclusion and exclusion. Hence find the number of

derangements of symbols 1,2,3,…..,n .

-4-

1B. 1C.

Prove that the number of partitions 2n with exactly n parts equals the number of partitions of n. (4 + 3 + 3)

2A. Prove that a graph is eulerian if and only if every vertex of it is of even degree.

2B.

Show that in any graph G number of vertices of odd degree is even.

2C. (4 + 3 + 3)

3A.

Show that subgroup of a cyclic group is cyclic.

3B.

Show that every group of order less than or equal to 5 is abelian.

3C.

Show that every group of prime order is cyclic. (4 + 3 + 3)

4A.

Prove that in a lattice L,

4B.

If the meet operation is distributive over the join operation in a lattice, then

both meet and join are associative.

show that the join operation is also distributive over the meet operation.

4C. (4 + 3 + 3)

-5-

Contd…..

5A. 5B.

5C.

Symbolise the following using predicates: (i)

All men are giants

(ii)

x is the father of y

(iii)

All elephants are grey.

Show that x

P(x)

Q(x)

x Q x .

x P x

(4 + 3 + 3) 6A.

State and prove Lagrange’s theorem.

6B.

Construct a grammar for the language L

6C.

Show that every tree has atleast two pendant vertices.

a n bn cn | n

1

( 4 + 3 + 3)

7A.

Let (A, ) be a distributive lattice. If a a

x

a

y & a

x

a

y for all

A show that x = y.

a) 7B. 7C.

Show that and

x M(x) follows logically from the premises (x) H(x)

M(x)

x H(x) . (4 + 3 + 3) *****

-6-

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