**MODEL PAPER**

**B.Tech (CSE) Degree Examination**

**3rd Sem**

**discrete Mathematical Structures I**

Effective from the admitted batch of 2004-2005

Time: 3 hrs

Max Marks: 70

First Question is Compulsory

Answer any four from the remaining questions

All Questions carry equal marks

Answer all parts of any question at one place

1. Answer the following

a) Write the elements of the set P(P(P(φ))) where P(A) denotes the power set of the set A and φ denotes the empty set.

b) Write the following statement in predicate calculus:

There is a brother for every person.

c) How many ways can 12 people have their birthdays in different calendar months?

d) Find the number of divisors of 400.

e) Write the characteristic equation of S_{k}-7S_{k-2}+6S_{k-3}=0.

f) Write the adjacency matrix of the following digraph.

—–DIAGRAM—–

g) Draw all possible binary trees with three nodes.

2. a) Check whether ((P→ Q)→ R)→((P→ Q)→(P→ R)) is a tautology.

b) Write the following sentences into predicate logic statements and verify the conclusion.

All trees are graphs.

Some graphs are trees.

AND – OR graph is a graph.

MST is a tree.

Therefore, MST is a graph.

3. a) Find the nuniber of integer solutions to the equation

x_{1} + x_{2} +x_{3} + x_{4} + x_{5} = 20 where x_{1} ≥ 3, x_{2} ≥ 2, x_{3} ≥ 4. x_{4} ≥ 6 and x_{5} ≥ 0.

b) A simple code is made by permuting the letters of the alphabet of 26 letters with every letter being replaced by a distinct letter. How many different codes can be made in this way?

4. a) Find the number of ways of placing 20 similar balls into 6 numbered boxes so that the first box contains any number of balls between 1 and 5 inclusive and the other 5 boxes must contain 2 or more balls each.

b) Solve a_{n} – 6a_{n-1}+12a_{n-2} – 8a_{n-3} – 0 by generating functions for n ≥ 3.

5. a) Find the transitive closure of the digraph whose adjacency matrix is

0 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0

b) Prove that a graph G is a tree if and only if G has no cycles and the number of the edges of G is one less than the number of vertices of G.

6. a) Write Kruskal’s algorithm for finding the minimum spanning tree of a graph

b) Find the minimum spanning tree of the graph given by the adjacency matrix

0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 0 0 1 0 0

7. a) State and prove the Euler’s formula for planar graphs.

b) How many different Hamiltonian cycles are there in a complete graph of n vertices? Justify your answer.

8. a) What is graph coloring? State the four-color theorem.

b) Prove that every simple planar graph is 5-colorble.