LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 31

M.C.A. DEGREE EXAMINATION – COMPUTER APPLICATION

FIRST SEMESTER – November 2008

MT 1902 – MATHEMATICS FOR COMPUTER APPLICATIONS

Date : 11-11-08 Dept. No. Max. : 100 Marks

Time : 1:00 – 4:00

Part A (Answer ALL questions) 2 x 10 = 20

Define Lattice homomorphism between two lattices.
With usual notations prove that (i)(ii) .
Define context free grammar.
What is the difference between deterministic finite automata and non-deterministic finite automata?
Let G = (N, T, P, S), where N = {S}, T = {a}, P: {S → SS, S → a}. Check whether G is ambiguous or unambiguous.
Give a deterministic finite automata accepting the set of all strings over {0, 1} containing 3 consecutive 0’s.
If R and S be two relations defined by and , then find

RS, RR and R.

Let and ,. Write the matrix of

of R and sketch its graph.

Define ring with an example.
State Kuratowski’s theorem.

Part B (Answer ALL questions) 5 x 8 = 40

(a) Show that De Morgan’s laws given by and hold in a

complemented, distributive lattice.

(OR)

(b) Let be a lattice. For any prove the following distributive inequalities:

) and .

(a) Show that L(G) = is accepted by the grammar G = (N, T, P, S) where N = {S,A} T = {a, b}, P consists of the following productions: S → aSA, S → aZA, Z → bZB, Z → bB, BA → AB, AB → Ab, bB → bb, bA→ ba.

(OR)

(b) Let the grammar G = ({S,A}, {a, b}, P, S) where P consists of S →aAS, S → a, A → SbA , A → SS, A → ba. For the string aabbaa find a

(i) leftmost derivation

(ii) rightmost derivation

(iii) derivation tree.

(a) (i) Define deterministic finite state automata.

(ii) Draw the state diagram for the deterministic finite state automata, M = where Q =, Σ ={a, b}, F = and δ is defined as follows:

δ	a	b

Check whether the string bbabab is accepted by M. (3+5)

(OR)

(b) Given an non-deterministic finite automaton which accepts L. Prove that there exists a deterministic finite automaton that accepts L.

(a) (i) Write short on Hasse diagram.

(ii) Let and relation be such that if x divides y. Draw the

Hasse diagram of . (4+4)

(OR)

(b) (i) Show that n³+2n is divisible by 3 using principle of mathematical induction.

(ii) If the permutations of the elements of {1,2,3,4,5} be given by

, then find

α ^-1^-1. (4+4)

(a) Prove that there is a one- to-one correspondence between any two left cosets of H in G.

(OR)

(b) (i) If G is a graph in which the degree of every vertex is atlest two, then prove that G

contains a cycle.

(ii) Prove that the kernel of a homomorphism g from a group to is a subgroup

of . (4+4)

Part C (Answer ANY TWO questions) 2 x 20 = 40

16.(a) Let G be (p,q)graph, then prove that the following statements are equivalent:

(i) G is a tree. (ii) Every two vertices of G are joined by a unique path (iii) G is connected

and (iv) G is acyclic and p = q+1.

(b) Let H be a subgroup of G. Then prove that any two left cosets of H in G are either

identical or have no element in common. (14+6)

(a) Let be a Boolean Algebra. Define the operations + and · on the elements of B by,

. Show that is a boolean ring with identity 1.

(b) Prove that every chain is a distributive lattice. (15+5)

(a) If G = (N, T, P, S) where N = {S, A,B}, T = {a,b}, and P consists of the following rules:

S → aB, S → bA, A → a, A → aS, A → bAA, B →b, B → bS, B → aBB. Then prove the following:

S w iff w consists of an equal number of a’s and b’s
A w iff w has one more a than it has b’s.
B w iff w has one more b than if has a’s

(b) State and prove pumping lemma. (10+10)

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