III BA, B.Sc. Mathematics Paper IV (Elective-2) - Acharya Nagarjuna ...

III B.A./B.Sc. Mathematics10Paper IV (Elective -2) - CurriculumACHARYA NAGARJUNA UNIVERSITYB.A / B.Sc. DEGREE EXAMINATION, THEORY MODEL PAPER(Examination at the end of third year, for 2010 - 2011 and onwards)MATHEMATICS PAPER - IV (ELECTIVE - 2)MODERN APPLIED ALGEBRATime : 3 Hours Max. Marks : 1001. State Peano axioms.SECTION - A (6 X 6 = 36 Marks)Answer any SIX questions. Each question carries 6 marks2. Show that the relation m< n means mn | (meaning that m is a divisor of n) is a partial ordering of theset of all positive integers.3. Explain utilities problem.4. If a graph (connected or disconnected) has exactly two vertices of odd degree, prove that theremust be a path joining these two vertices.5. Draw the state diagram for the following machine.Present v ξstate 0 1 0 11 1 2 0 02 2 3 0 03 3 4 0 04 4 1 0 16. Write ALGOL expressions for i)− b +b2 − 4ac2a⎛ x⎞ii) sin 3⎜ ⎟ iii) a b+ cq⎝ y⎠4d7. Define Boolean algebra.8. Prove that in any Boolean algebra, a∧ x =0 and a∨ x =1 imply x = a′ .9.(a)SECTION - B (4 X 16 = 64 Marks)Answer ALL questions. Each question carries 16 marksProve that a function is left invertible iff it is one one.n(b) Prove by induction that Σ k2nn ( + 1)( 2n+1) =where n is any positive integer.k = 16OR10.(a) Prove that an equivalence relation on a set S gives rise to a partition on S.(b)11.(a)If ρ and σ are reflexive and symmetric relations on a set S, then show that the following areequivalent.i) ρσ is symmetric ii) ρσ = σ ρ iii) ρσ = σ ∨ ρ.Prove that a connected graph G is an Euler graph iff it can be decomposed into circuits.