LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
SIXTH SEMESTER – APRIL 2008
MT 6601 – APPLIED ALGEBRA
Date : 21/04/2008 Dept. No. Max. : 100 Marks
Time : 9:00 – 12:00
PART – A
Answer ALL questions: (10 x 2 = 20)
- Construct the truth table for the statement formula Pù Q.
- What is meant by well formed formula.
- When is a statement A said to tautologically imply a statement B?
- Define conditional statement.
- State Zorn’s lemma
- Define Boolean Algebra and give an example of it.
- Define Mealy and More automaton.
- Give an example of a semi group which is not a monoid.
- Define anti homomorphism and anti isomorphism of semi groups.
- Define automata homomorphism.
PART – B
Answer any FIVE questions. (5 x 8 = 40)
- Show that
- ( P®(Q®R)ÛP®(ùQÚR) Û(PÙQ) ®R
(ii) ( ù PÙ (ù QÙ R) Ú(QÙR) Ú(PÙR) ÛR
- Show that
((PÚQ) Ùù ( ù PÙ(ùQÚùR)) Ú( ùPÙùR) is a tautology.
- Obtain a conjunctive normal form of each of the formula as given below
(i) (PÙ(P ®Q) (ii) ù (PÚQ) D (PÙQ)
- Obtain the product – of –sums canonical forms of the following formulas.
(i) (PÙQ) Ú ( ù PÚQ) Ú(PÙù Q)
(ii) (PÙQ) Ú ( ù PÙ QÙR)
- Let be a lattice ordered set. If Π; show that (L,Π, ) is an algebraic lattice.
- State and prove Representation theorem for Boolean Algebra.
- For any semigroup (S,o), Show that there exists a set N such that (S,o) is embeddable in (NN, o).
- Show that for any monoid (S,o) there exists a semi automation whose monoid is isomorphic to (S,o).
PART – C
Answer any TWO questions. (2 x 20 = 40)
- a) Show that
(i) ù (PÙQ) ® ( ù PÚ( ù PÚQ) Û (ù PÚQ)
(ii) (PÙQ) Ù ( ù PÙ ( ù PÙQ) Û (ù PÙQ)
- b) Show that the truth values of the following formulas are independent of their
(i) (PÙ ( P® Q) ® Q
(ii) (P® Q) D( ù PÚQ)
- a) Obtain the principal disjunctive normal forms of
(i) ( ù PÚQ)
(ii) (PÙQ) Ú( ù PÙR) Ú( QÙR)
- b) Show that a modular lattice is distributive if and only if none of its sub lattices is isomorphic to the diamond lattice .
- a) Explain marriage semi automation.
- b) Let ~1 and ~2 be two congruence relations on (S,o). Show that ~1, C ~2 if and only if is an epimorphism of (S/~1, o) onto (S/~2, o).
- c) For any f show that there exists a semi group F which is free on B.
- a) For two automation if show that is a homomorphic image of .
- b) If the automation is a homomorphic image of , show that is a homomorphic image of .
- c) Explain parallel composition and series composition of two automata and .
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