LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

XZ 30

M.C.A. DEGREE EXAMINATION – COMPUTER APPLICATION

FIRST SEMESTER – APRIL 2008

          MT 1902 / CA 1804 – MATHEMATICS FOR COMPUTER APPLICATIONS

 

 

 

Date : 05/05/2008            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

SECTION A

Answer ALL the questions.                                                                         (10 x 2 = 20)

 

1. Define least upper bound of a poset.
2. Define a Lattice.
3. What are the logic operators?
4. Construct a phrase structure grammar for the language .
5. Define context-sensitive language.
6. For a DFA ,

show that the string 011011 is in L(M)

7. State the Pigeon hole principle.
8. Draw the Hasse diagram for the divisors of 32.
9. Define a bipartite graph with an example.
10. Prove that every cyclic group is abelian.

 

SECTION B

Answer ALL the questions.                                                                         (5 x 8 = 40)

 

11. (a) Prove that the complement of any element ‘a’ of a Boolean algebra is uniquely determined.           Prove also that the map  is an anti – automorphism of period  2 and  satisfies                 (a Ú b)¢ = a¢ Ù b¢, (a Ù b)¢ = a¢ Ú b¢, a¢¢ = a.

(or)

(b) Discuss ‘negation’ and explain a method of constructing the truth table for P Ú ùQ and (P Ú Q) Ú ùP

 

12. (a) Write a short note on principal conjunctive normal form and construct an equivalent formula             for ù.

(or)

(b) For a grammar  where P consists of the following production:

Then show that.

 

13. (a) Let L be a set accepted by a non-deterministic finite automaton. Then prove that there exists a deterministic finite automaton that accepts L

(or)

(b) (i) Construct an equivalent deterministic automaton for a given non-deterministic automatonwhere .

(ii) If R and S are equivalence relations on the set X, prove that R Ç S is also an equivalence relation on X.

 

14. (a) Prove that the equivalence relation ~ defined on the set A decomposes the set A into mutually disjoint equivalence classes.

(or)

(b) (i) A computer password consists of a letter of the alphabet followed by 3 or 4 digits. Find the total number of passwords that can be formed and the number of passwords in which no digit repeats.

(ii) Find the minimum number of students in a class to be sure that four out of them are born in the same month.

 

15. (a) (i) Prove that a subgroup N of a group G is a normal subgroup of G iff the product of two left cosets of N in G is again a left coset N in G.

(ii) Define ring with an example.

(or)

(b) Prove that the following statements are equivalent for a connected graph G.

• G is Eulerian
• Every point of G has even degree
• The set of edges of G can be partitioned into cycles.

 

SECTION C

Answer any TWO questions.                                                                       (2 x 20 = 40)

 

16. (a) Explain conditional and bi-conditional connectives with an example.

(b) Define a Non – Deterministic Finite automata.

(c) For the non deterministic finite automaton,

 

give the transition table and show that 0100110 is in L (M).

 (10 + 2 + 8)

 

17. (a) State and prove pumping lemma for regular sets.

(b) List any four applications of pumping lemma.

(c) Prove that if  and  be one-to-one onto functions, then  is also one-to-one onto and .

(10 + 4 + 6)

 

18. (a) Show that in a graph G, any u – v walk contains a u – v path.

(b) Prove that a closed walk of odd length contains a cycle.

(c) State and prove Lagrange theorem.                                                            (4 + 4 + 12)

 

 

 

 

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

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

RS, RR and R.

8. Let and ,. Write the matrix of

of R and sketch its graph.

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

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

11. (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 .

12. (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.

13. (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.

14. (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)

 

 

15. (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)

 

17. (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)

18. (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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