LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
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
Answer ALL the questions. (10 x 2 = 20)
- Define least upper bound of a poset.
- Define a Lattice.
- What are the logic operators?
- Construct a phrase structure grammar for the language .
- Define context-sensitive language.
- For a DFA ,
show that the string 011011 is in L(M)
- State the Pigeon hole principle.
- Draw the Hasse diagram for the divisors of 32.
- Define a bipartite graph with an example.
- Prove that every cyclic group is abelian.
Answer ALL the questions. (5 x 8 = 40)
- (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.
(b) Discuss ‘negation’ and explain a method of constructing the truth table for P Ú ùQ and (P Ú Q) Ú ùP
- (a) Write a short note on principal conjunctive normal form and construct an equivalent formula for ù.
(b) For a grammar where P consists of the following production:
Then show that.
- (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
(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.
- (a) Prove that the equivalence relation ~ defined on the set A decomposes the set A into mutually disjoint equivalence classes.
(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.
- (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.
(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.
Answer any TWO questions. (2 x 20 = 40)
- (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)
- (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)
- (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)