Assignment for M.Sc. (Maths) Part II (Sem. - III) Distance Mode.

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1 Assignment for M.Sc. (Maths) Part II (Sem. - III) Distance Mode. Last Date for Assignment Submission: - 10 th October, th Mar These assignments are to be submitted only by those students who have registered for the Course M.Sc Part II (SEM-III) 2016 through Distance mode. Sr. No. Descrition Page No. 01 Instructions for Submission of Assignments General instructions 3 03 Cover age format 4 05 M.Sc(Maths) Part I I (Sem-III) Distance Mode Assignments 5-7

2 Assignment Questions for M.Sc. (Mathematics) Part II (Sem. III) Oct./Nov Instructions for Assignment Submission Please note following instructions for submission of Assignments: 1. These assignments should be submitted only by those students who have registered for M.Sc. (Maths) Part I (Sem. I) Distance mode Fresh and Reeaters, the examinations to be held in Oct./Nov For each subject s assignment the maximum marks obtainable are All Questions are comulsory for each course/aer/subject, assignment should be Hand Written on a searate sheet. 4. Only Blue Colored Ink Pen is to be used for Assignment Writing. 5. Only A4 / Journal/Assignment aer should be used for the Assignment Writing. 6. A searate set should be made for each subject. 7. A cover age as er the format given below on age number 4 should be attached on the to of the set for each subject. 8. Finally for a articular semester, one file should be made for all subjects. 9. Submit the Assignments to : Centre for Distance Education, Shivaji University, Kolhaur. Pin Telehone: - (0231) , It is the student s resonsibility to ensure that the assignments reach the centre on or before the due date. No excuses of any kind for late or non-submission of assignments will be entertained. If a student is unable to submit the assignment(s) in erson, the student may at his / her own risk submit the assignment(s) through an acquaintance, fellow student or by courier. If assignments are sent by Seed Post / courier, at the to of the enveloe the student should clearly write in BOLD letters ASSIGNMENT FOR M.Sc.(Maths) Part II (Sem.III) Oct./Nov DISTANCE MODE

3 General Instructions a. Please note that the student has to obtain at least 12 marks out of 30 marks in internal assignments and 36 marks out of 90 marks in university examinations. b. Students are advised that imrovement in assignment marks is not ermitted at a later stage once the student gets the minimum assing marks or more (i.e. 12+ marks). Hence the students are advised to try to score the maximum at the first attemt. c. Assignments should not be coied, should be clear, legible, well resented. d. Illustrate your answer by giving suitable examles. e. Draw grahs or diagrams wherever necessary. f. Students are advised that in case two or more students assignments are too similar in content, nature, the study center Co-Ordinator would at his / her discretion decide on the quantum of marks to be awarded, irresective of how good the submitted assignments are. It is more than likely that the minimum ossible marks (if any) may be awarded to all such involved assignments. g. Students are also advised to quote sources (if any) of data, facts, sketches, drawings etc in their assignments. h. In case of any query contact Coordinator Centre for Distance Education, Shivaji University, Kolhaur. Telehone: - (0231) , cde_multi@unishivaji.ac.in Students should see their Namelist, PRN and Seat Nos./Hall Tickets on the following website : Website : online.shivajiuniversity.in (Download Hall Ticket for Distance Education otion - referably through Google Chrome.) Last Date for Assignment Submission:- 10 th October, th Mar. 2016

4 M.Sc. (Maths.) Part II Sem. III Oct./Nov Distance Mode Assignment for the Subject of Paer Number: - Subject Code:- 1. Name of the Candidate :- 2. Name of the Study Centre 3. Address: - _ Pin:- Mobile No: - 4. Exam Seat Number: - PRN Number : 5. Course: - M.Sc. (Maths) Part II (Semester III) Distance Mode. 6. Date of Submission of Assignments: - 7. Signature of Student: - 8. Marks obtained out of 30:- 9. Signature of Evaluator of Assignment: -

5 M.Sc. (Maths) Part II (Sem. III) Assignment Questions Sub. : Functional Analysis (63303) Q.1 Let denote a linear sace of set of all bounded sequences of scalars with norm 1/ x xn (1 ) n1. Then show that is Banach sace. Q.2 Prove that on finite dimensional sace all norms are equivalent. Q.3 Prove that * q, where 1 1 1, 1 q. Q.4 State and rove Pythagorean theorem Q.5 Prove that orthonormal set in a Hilbert sace is linearly indeendent. Q.6 Prove that an oerator T on finite dimensional Hilber sace H is normal if and only if its adjoint is olynomial in T. Sub. : Advanced Discrete Mathematics (63304) Q.1 a. Let G be a k-regular grah, where k is odd number. Prove that the number of edges in G is a multile of k. b. Let G be a simle grah with at least two vertices. Prove that G must contain two or more vertices of the same degree.. Q.2 a. Prove that if given a set of any seven distinct integers, there must exist two integers in this set whose sum or difference is multile of 10. b. Prove that a grah G is connected if and only if it has a sanning tree. Q.3 a. Find the numeric function corresonding to the generating function A(z) = () b. Find the generating function corresonding to the numeric function (0, 1, 2,.., r,.. ). Q.4 a. Find the number of integers between 1 and 1000 inclusive which are relatively rime to 3,5 and 7. b. Solve a r + 5 a r a r-2 = 3r 2r + 1 Q.5 a. Let L be a finite distributive lattice. Prove that every element a in L can be written uniquely as the join of irredundant join irreducible elements. b. Show that the following are equivalent in a Boolean algebra i) a + b = a ii) a x b = a iii) a + b = 1 iv) a x b = 0

6 Sub. : Number Theory (63307) 4 4 Q.1 Show that if a and b are both odd integers, then 16 a b 2. Q.2 Prove that the number 3 is irrational. Q.3 We have an unknown number of coins. If you make 77 strings of them, you are 50 coins short; but if you make 78 strings, it is exact. How many coins are there? Q.4 State and rove Chinese Remainder Theorem. Q.5 For n 1, rove that the sum of the ositive integers less than n and relatively rime to n is 1 ( ) 2 n n. Q.6 If is an odd rime, rove that there exists a rimitive root r of such that r 1(mod ). 1 2 Sub. : Oeration Research - I (63316) Q.1 If x is a feasible solution to the rimal and w is a feasible solution to the dual such that c.x = b.w then show that x is an otimal solution to the rimal and w is an otimal solution to the dual. Q.2 Use revised simlex method to solve linear rogramming roblem. Maximize z = x 1 + 2x 2 subject to the constraints x 1 + x 2 3, x 1 + 2x 2 5 3x 1 + 2x 2 6 x 1, x 2 0 Q.3 A ositive quantity c is to be divided into 3 arts in such a way that the roduct of these three arts is to be a maximum. Obtain the otimal subdivision. Q.4 What is integer rogramming roblem? Exlain Branch and bound method used to solve integer rogramming roblem. Q.5 Otimize z = 4x x x 3 2 4x 1 x 2. Subject to x 1 + x 2 + x 3 = 15, 2x 1 x 2 + 2x 3 = 20 and x 1,x 2, x 3 0. Q.6 State and rove Kuhn Tucker necessary and sufficient conditions for otimization of nonlinear rogramming roblems.

7 Sub. : Fuzzy Mathematics (63314) Q.1 a. If A (x) =, B (x) = where x X, X = [0, 10]. Determine the mathematical formula for the following fuzzy sets (i) A B (ii) A B (iii) A B (iv) A B b. Show that a fuzzy set A is convex if and only if A ( x1 + (1 - ) x2) min {A (x1), A (x2)}. x, x 2 X. c. Let C : [0, 1] [0, 1] be a function. Prove that C is an involutive fuzzy comlement if and only if there exists a continuous function g : [0, 1] 1 R such that g (0) = 0, g is strictly increasing and c (a) = g -1 (g (1) g (a)) Q.2 a. Show that U w (a, C w (a)) = 1, a [0, 1] w > 0 where U w and C w denote Yager Union and comlement resectively. b. Show that the function g defined by g (a) = a, 0 a is an increasing < a 1 generator and find the fuzzy comlement, t- norm and t conorm generated by g. c. Let U w and C w be Yager fuzzy union and fuzzy comlement. Find the dual fuzzy intersection of U w with resect to C w. Q.3 a. If A, B, C are closed intervals then rove that (i) A + B = B + A, A. B = B. A (ii) A. (B + C) A. B + A. C b. Let A, B be two fuzzy numbers given by A (x) = otherwise B (x) =, 2 < x 0 B (x) = 0, 0 < x < 2,, 2 < x 4, 0 < x < 6, other wise 0 Find A + B, A B, A. B and A/B.,