First year syllabus under Autonomy First year syllabus under Autonomy

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First year syllabus under Autonomy First year syllabus under Autonomy Shri Vile Parle

COLLEGE OF COMMERCE AND ECONOMICS (AUTONOMOUS) NAAC Reaccredited A grade, CGPA: 3.

University of Mumbai 2016-17 Affiliated to the UNIVERSITY OF MUMBAI Program: B.Sc. Course: F.

1 First year syllabus under Autonomy First year syllabus under Autonomy Shri Vile Parle Kelavani Mandal s MITHIBAI COLLEGE OF ARTS, CHAUHAN INSTITUTE OF SCIENCE & AMRUTBEN JIVANLAL COLLEGE OF COMMERCE AND ECONOMICS (AUTONOMOUS) NAAC Reaccredited A grade, CGPA: 3.57, Granted under FIST-DST & Star College Scheme of DBT, Government of India Best College, University of Mumbai Affiliated to the UNIVERSITY OF MUMBAI Program: B.Sc. Course: F.Y.B.Sc. Mathematics (USMT) Credit Based Semester and Grading System (CBGS) with effect from the academic year

2 PREAMBLE With the introduction of Credit Based Semester & Grading System (CBSGS) and continuous evaluation consisting of components of Internal Assessment & External Assessment by the esteemed University from the academic year at F.Y.B.Sc.level, the earlier existing syllabus of F.Y.B.Sc. Mathematics was restructured according to the CBSGS pattern for implementation from Likewise the existing syllabi of S.Y.B.Sc. and T.Y. B.Sc. Mathematics were restructured as per the CBSGS pattern for their implementation from and respectively. Some of the modules of the earlier syllabus of F.Y.B.Sc. have been upgraded with the new modules in order to make the learners aware about the recent developments in various branches of Mathematics. It has applications to Computer Science, Social Science, Engineering & Technology, Operation Research. Syllabus covers the prerequisites of Calculus and Discrete Mathematics. Calculus covers the topics such as Real Numbers, Sequences and Series, Limits and Continuity, Differentiation and its Applications. Discrete Mathematics covers the topics such as Set Theory, Relation, Function, Natural numbers, Equivalence of Sets, Permutations, Counting Techniques, Integers, Complex Numbers, Polynomial with real coefficients. Both the courses of theory (Semester-I & Semester-II) are compulsory for the students offering Mathematics as a single major subject. These courses are:- 1. USMT101and USMB USMT201 and USMT202 I am thankful to co-conveners and all the members of our sub-committees for their great efforts and for timely submission of the draft syllabus.

3 COURSE CODE USMT101 F. Y. B. Sc. MATHEMATICS SEMESTER-I TITLE Calculus - I CREDITS AND LECTURES/SEM 3 credits (45 Lectures) Unit I Real Number System 15 lectures Unit II Sequences 15 lectures Unit III Graphs, Limits and Continuity 15 lectures USMT102 Discrete Mathematics - I 3 credits Unit I Unit II Unit III Integers and Divisibility Elementary Set Theory, Functions, Binary Operation, Equivalence Relation, Partition of a set. Euler s phi-function, Fermat s Theorem, Wilson s Theorem(Applications Only), Congruence Relation, Polynomials with real coefficients (45 Lectures) N.B.- (I) Each theory lecture shall be of 48 minutes duration. (II) Each tutorial shall be of 48 minutes duration.

4 F.Y.B.Sc. Mathematics: USMT-101 (Calculus I) Learning Objectives: To learn basic concepts related to Real Numbers and Real Sequences To learn the definition of convergence as applied to sequences. To learn various results and definitions related to Real Sequences and their convergence. To familiarise students with Graphs of standard functions To learn properties and theorems related to the limit and continuity of functions. Learning Outcomes: Students should be able to- Use results and write proofs related to basic properties of real numbers and sequences. Use results to solve problems and write rigorous proofs related to Real Sequences. Understand concepts related to functions, limits and continuity and awareness of the applications to various other disciplines. Use results and theorems to solve problems and write rigorous proofs.

5 USMT101: DETAILED SYLLABUS Course Code USMT101 UNIT I UNIT II UNIT III Title Calculus - I Real Number System 1.1. Introduction to Real Numbers 1.2. Order properties 1.3. Absolute Value function & its Properties 1.4. A.M. G.M inequality 1.5. Cauchy Schwartz inequality 1.6. Hausdroff s Property with proof 1.7. lub axiom, Supremum/infimum 1.8. Archimedean property with proof Sequences 2.1. Introduction & Definition 2.2. Types of sequences 2.3. definition of Convergence & examples 2.4. Algebra of Convergent Sequences 2.5. Sandwich Theorem with Proof & examples 2.6 definition of Monotone sequence 2.7. Monotone Convergence Theorem with proof & examples 2.8 definition of Subsequence & example 2.9 definition of Cauchy sequences, examples & results Limits and Continuity 3.1. Graphs of standard functions 3.2. Analysis of function using graph 3.3. Definition of Limit of a function 3.4. Algebra of Limit of a function 3.5. Sandwich Theorem for functions with Proof & examples 3.6. Definition of Continuity of function 3.7 Sequential continuity & results 3.8 Discontinuity & examples Lectures/ Semester 3 Credits (45 Lectures) 15 Lectures 15 Lectures 15 Lectures

6 Assignments (Tutorials) 1. Application based examples of Archimedean property, intervals, neighbourhoods. 2. Infimum and Supremum of sets. 3. Calculating limits of sequence. 4. Cauchy sequence, monotone sequence. 5. Limit of a function and Sandwich theorem. 6. Continuous and discontinuous functions. References 1. R. R. Goldberg, Methods of Real Analysis, Oxford and IBH, Binmore, Mathematical Analysis, Cambridge University Press, T. M. Apostol, Calculus Vol I, Wiley & Sons (Asia) Pte. Ltd. 4. Courant and John, A Introduction to Calculus and Analysis, Springer. 5. Ajit and Kumaresan, A Basic Course in Real Analysis, CRC Press, James Stewart, Calculus, Third Edition, Brooks/cole Publishing Company, Ghorpade Limaye, A Course in Calculus and Real Analysis, Springer International Ltd, 2000.

7 F. Y. B. Sc. Mathematics: USMT-102 (Discrete Mathematics I) Learning objectives: Discrete Mathematics is an important branch of Mathematics with applications to Computer Science, Social Science, Engineering & Technology and Operation Research, etc. At degree level, a student taking Mathematics is not tested for his/her skill of taking derivatives and integrals alone. Instead he is exposed to abstract Mathematics as well. In abstract mathematics, one defines certain concepts and then deduces the entire theory from those. Hence the basic knowledge of logic will be very much helpful in developing the subject. Learning Outcomes: Students should be able to- Understand the results of Set theory which are used to study relations, functions and natural numbers. Equivalence of sets introduce the concept of finite/infinite sets. Understand the counting techniques which gives the tools to handle finite sets more efficiently. Understand the properties of integers which are used to introduce the ides of congruence and linear congruence. Understand the results in complex numbers which are introduced as the algebraic set of order pairs. The properties of complex numbers help the students to understand the beauty of results in polynomials.

8 USMT-102: DETAILED SYLLABUS Course Code Title Lectures/ Semester USMT-102 Discrete Mathematics - I 3 Credits (45 lectures) UNIT I UNIT II Natural Numbers, Integers & Divisibility 1.1. Introduction & Properties of N Well ordering principle Theorem & Principles of induction 1.2 Introduction & Properties of Z Divisibility in integers Division Algorithm 1.3 G.C.D & L.C.M Properties of G.C.D G.C.D Theorem Euclidean Algorithm & Applications Euclid s lemma 1.4 Prime & results Fundamental theorem of Arithmetic Elementary Set Theory, Functions, Binary Operation, Equivalence Relation, Partition of a set 2.1 Introduction to elementary Set Theory Definition and examples Operations on sets 15 Lectures 2.2 Functions and Binary Operation Definition and examples of function Types of functions Operations on functions Properties of Binary Operations and Examples 15 Lectures 2.3 Equivalence Relation and Partition of a set Reflexive, Symmetric and transitive relation and examples Equivalence classes Theorems on equivalence relations Partitions of a set with examples Relation between Partition and Equivalence relation

9 UNIT III Euler s phi-function, Fermat s Theorem, Wilson s Theorem, Congruence Relation, Polynomials with real coefficients 3.1. Congruence Relation Properties of congruence relation Examples based on these 3.2. Introduction to Number Theory Euler s phi-function and properties Euler s theorem (Statement ONLY) and examples based on it Fermat s Little Theorem : Proof and Examples Wilson s Theorem (Statement ONLY) and examples based on it 15 Lectures 3.3. Polynomials with real coefficients Division algorithm(statement ONLY) GCD of polynomials Fundamental Theorem of Algebra(Statement ONLY) Remainder Theorem Factor Theorem Rational Root Test Multiplicity of a root

10 Assignments (Tutorials) 1. Mathematical Induction Principle 2. Division algorithm and GCD, LCM of integers 3. Function and Binary Operation 4. Equivalence Relation and Partition 5. Congruence Relation and Application of Euler s, Fermat s and Wilson s Theorem 6. Roots of a polynomial and multiplicity Reference Books 1. David M. Burton, Elementary Number Theory, Seventh Edition, McGraw Hill Education (India) Private Ltd. 2. Norman L. Biggs, Discrete Mathematics, Revised Edition, Clarendon Press, Oxford I. Niven and S. Zuckerman, Introduction to the theory of numbers, Third Edition, Wiley Eastern, New Delhi, G. Birkoff and S. Maclane, A Survey of Modern Algebra, Third Edition, Mac Millan, New York, N. S. Gopalkrishnan, University Algebra, New Age International Ltd, Reprint I.N. Herstein, Topics in Algebra, John Wiley, P. B. Bhattacharya S. K. Jain and S. R. Nagpaul, Basic Abstract Algebra, New Age International, Kenneth Rosen, Discrete Mathematics and its applications, Mc-Graw Hill International Edition, Mathematics Series.

11 F. Y. B. Sc. MATHEMATICS SEMESTER-II COURSE CODE USMT201 Unit I Unit II TITLE Calculus - II Continuous functions and Differentiation Application of differentiation CREDITS AND LECTURES/SEM 3 credits (45 Lectures) 15 lectures 15 lectures Unit III First order First degree Differential equations 15 lectures USMT202 Discrete Mathematics - II 3 credits Unit I Unit II Unit III Counting Techniques Permutations Complex Numbers and Recurrence Relation (45 Lectures) N.B.- (I) Each theory lecture shall be of 48 minutes duration. (II) Each tutorial shall be of 48 minutes duration.

12 F.Y.B.Sc. Mathematics: USMT-201 (Calculus II) Learning Objectives: To learn theorems and properties related to continuous functions To study differentiability of a function at a point. To understand maxima and minima of a function To learn process to draw graphs of differentiable functions Introduction to Differential Equations & types of Solution, Homogenous, Exact differential equation & procedure to solve. Equations reducible to the exact form by using integrating factors (four rules) Linear differential equations & differential equations reducible to normal form (Bernoulli s equation) Learning Outcomes: Students should be able to- Check differentiability of a function at a point. Find maximum and minimum values of a function. Draw graph differentiable functions and use L hospital s rule to solve limit problems. Solve problems using theorems based on differentiation. Understand definition of exact differential equations & will be able to solve exact differential equations Understand the concept of an Integrating factor & will be able to solve example based on it. Students will be able to identify linear differential equation & can solve the examples based on it.

13 Course Code USMT201 UNIT I UNIT II UNIT III USMT201: DETAILED SYLLABUS Title Calculus - II Continuous functions and Differentiation 1.1. Introduction 1.2. Algebra of continuous functions 1.3. Properties of Continuous functions 1.4. Intermediate Value theorem and applications Definition of Differentiation of real valued function of one variable & examples 1.6. Higher order derivatives 1.7. n th order derivative formula for Standard functions and examples 1.8. Leibnitz rule for derivatives (with Proof) & examples Application of differentiation 2.1 Local maximum, minimum, necessary condition, stationary points, second derivative test, examples 2.2 Graphing of functions using first and second derivatives, concave and convex functions, points of inflection 2.3 Rolle s Theorem, Lagrange s and Cauchy s Mean value theorem 2.4 Monotone functions and examples 2.5 L hospital s rule and examples 2.6 Taylor s Theorem and polynomial and examples First order First degree Differential equations 3.1 Definition of a differential equation, order, degree, ordinary differential equation 3.2 Review of solution of homogeneous and nonhomogeneous differential equations of first order and first degree 3.3. General Solution of Exact equations of first order and first degree. Non-exact equations. Rules for finding integrating factors (without proof) for non-exact equations. 3.4 Linear and reducible to linear equations 3.5 Applications to orthogonal trajectories, population growth and decay Lectures/ Semester 3 Credits (45 Lectures)

14 Assignments (Tutorials) 1. Continuous functions and Intermediate Value theorem 2. Higher order derivatives & Leibnitz rule 3. Graphing of functions & Mean value theorems 4. L hospital s rule & Taylor s Theorem 5. Exact differential equations and Non-exact differential equations. 6. Linear differential equations and its Applications References 1. R. R. Goldberg, Methods of Real Analysis, Oxford and IBH, Binmore, Mathematical Analysis, Cambridge University Press, T. M. Apostol, Calculus Vol I, Wiley & Sons (Asia) Pte. Ltd. 4. Courant and John, A Introduction to Calculus and Analysis, Springer. 5. Ajit and Kumaresan, A Basic Course in Real Analysis, CRC Press, James Stewart, Calculus, Third Edition, Brooks/cole Publishing Company, Ghorpade Limaye, A Course in Calculus and Real Analysis, Springer International Ltd, 2000.

15 F. Y. B. Sc. Mathematics: USMT-202 (Discrete Mathematics- II) Learning objectives: Discrete Mathematics is an important branch of Mathematics with applications to Computer Science, Social Science, Engineering & Technology and Operation Research, etc. At degree level, a student taking Mathematics is not tested for his/her skills of taking derivatives and integrals alone. Instead he is exposed to abstract Mathematics as well. In abstract mathematics, one defines certain concepts and then deduces the entire theory from those. Hence the basic knowledge of logic will be very much helpful in developing the subject. Learning Outcomes: Students should be able to- Understand various counting techniques which are used to handle problems on finite sets and apply in day-to-day life. Understand and perform different fundamental operations related to permutations and use it in higher courses. Understand the results in complex numbers which are introduced as the algebraic set of order pairs. The properties of complex numbers help the students to understand the beauty of results in polynomials. Understand to formulate real life problems using recurrence relation and to solve it

16 USMT-202: DETAILED SYLLABUS Course Code Title Lectures/ Semester USMT-202 Discrete Mathematics - II 3 Credits (45 lectures) UNIT I UNIT II UNIT III Counting Techniques 1.1 Pigeonhole Principle 1.2 Counting Principles 1.3 Principle of Inclusion and Exclusion 1.4 Binomials and Multinomials 1.5 Derangements 1.6 Applications of Stirling s Number of Second kind Permutations 2.1 Introduction to permutation 2.2 Composition of permutations 2.3 Cyclic permutation 2.4 Transposition 2.5 Inversion of permutation 2.6 Sign of a permutation, odd/even permutations 2.7 Decomposition of a permutation as a product of disjoint cycles Complex Numbers and Recurrence Relation 3.1 Complex Numbers : Introduction, Cartesian and polar form 3.2 Geometrical Interpretation using Argand s diagram 3.3 Algebra of Complex Numbers 3.4 De Moivre s theorem and its applications 3.5 n th roots of unity and n th roots of complex numbers 3.7 Introduction to linear recurrence relation and formation of real life problems. 3.8 Solution using back-tracking method 3.9 General methods to solve homogeneous linear recurrence relation with proof.

17 Assignments (Tutorials) 1. Pigeonhole & Counting Principle and Inclusion-Exclusion Principle 2. Binomial, Multinomial, Derangements and Stirling s Number 3. Permutations and its applications 4. Sign of a permutation, Even/odd permutation 5. Complex Numbers 6. Linear Recurrence relation Reference Books 1. Norman L. Biggs, Discrete Mathematics, Revised Edition, Clarendon Press, Oxford G. Birkoff and S. Maclane, A Survey of Modern Algebra, Third Edition, Mac Millan, New York, N. S. Gopalkrishnan, University Algebra, New Age International Ltd, Reprint I.N. Herstein, Topics in Algebra, John Wiley, P. B. Bhattacharya S. K. Jain and S. R. Nagpaul, Basic Abstract Algebra, New Age International, Kenneth Rosen, Discrete Mathematics and its applications, Mc-Graw Hill International Edition, Mathematics Series.

UNIVERSITY OF MUMBAI

Academic Council 25/05/2011 Item No. 4.25 UNIVERSITY OF MUMBAI SYLLABUS for the F.Y.B.A/B.Sc. Programme:B.A./B.Sc. Course: (Credit Based Semester and Grading System with effect from the academic year 2011