BSC (NM/CS) 2 YEAR

Minor Courses

(To be offered to the students of other departments)

Semester	Course Code	Name of Course	Credits
I	MAT-M-1	Calculus	6
II	MAT-M-2	Algebra and Ordinary Differential Equations	6
III	MAT-M-3	Partial Differential Equations and Numerical Analysis	6
IV	MAT-M-4	Number Theory and Group Theory	6
V	MAT-M-5	Mathematical Modelling	4
VI	MAT-M-6	Discrete Mathematics	4
VII	MAT-M-7	Special Functions and Integral Transformations	4
VIII	MAT-M-8	Advanced Calculus / Linear Algebra	4

The above Minor Courses will be offered only to those students who studied Mathematics in 10+2.

Skill Enhancement Compulsory Courses

(3 Credits)

Semester	Course Code	Name of Course
I	MAT-SEC-1	Discrete Mathematics
II	MAT-SEC-2	Working with Mathematical Softwares
IV	MAT-SEC-3	Programming in C

Multidisciplinary Courses

(3 Credits)

(To be offered to the students of other departments)

Semester	Course Code	Name of Course
I, II	MAT-IDC	Algebra and Geometry
III	MAT-IDC-3	Matrices

These IDC courses will be offered only to those students who did not study Mathematics after matriculation.

Students are not permitted to repeat any course they have already completed for credit.

Ability Enhancement Compulsory Courses

A student is required to take two Ability Enhancement Courses (Language) of 2 credits per semester in Semesters I and II from the pool of AECC courses offered by the University.

Value Added Courses

One Value Added Course of 2 credits per semester in Semesters I, II and V shall be chosen from the pool of VAC courses offered by the University.

The department may also offer some of the following Value Added Courses (VACs). These courses are not open to students of B.Sc. (Mathematics) or B.Sc. (Mathematics and Computing). However, the VAC course on Number Theory may be opted by students of B.Sc. (Mathematics and Computing).

Course Code	Name of Course
MAT-VAC-1	Logic and Sets
MAT-VAC-2	LaTeX and HTML Theory-I
MAT-VAC-3	Discrete Mathematics
MAT-VAC-4	Matrices and Their Applications
MAT-VAC-5	Graph Theory
MAT-VAC-6	Probability and Statistics
MAT-VAC-7	Number Theory

Students are not permitted to repeat any course they have already completed for credit.

MAT-DSC-3: Metric Spaces

Credits: 6 (L=5; T=1; P=0)

Total Marks: 150 (Including Internal Assessment = 30)

Time Allowed for Examination: 3 Hours

Instructions for the Candidates and Paper Setters

Candidates will be required to attempt five questions out of nine questions carrying equal marks.

Question No. 1, covering the entire syllabus, will be compulsory.

There will be two questions from each unit, and students will be required to attempt one question from each unit.

Learning Outcome

This course introduces the fundamentals of Real Analysis and Metric Spaces.

Students will study convergence, limits, and continuity within metric spaces along with concepts such as completeness, compactness, and connectedness in general metric spaces, especially finite-dimensional Euclidean spaces.

The course also covers countable and uncountable sets, separable sets, perfect sets, equivalent metrics, homeomorphisms, and the Cantor set.

Unit-I

Real Number System and Sequences

• The Real Number System
• Least upper bound property of real numbers
• Archimedean property
• Sequences of real numbers
• Convergence of sequences
• Algebra of limits
• Squeeze Rule
• Bounded monotone sequences
• Monotone Subsequence Theorem
• Bolzano-Weierstrass Theorem
• Nested Interval Property
• Cauchy sequences
• Series convergence
• Comparison test
• Root test
• Ratio test for convergence of series
• Limit and continuity of real functions

Scope: Sections 1.1 – 1.3 and 1.5 of Textbook (A)

Unit-II

Metric Spaces and Topology

• Metric spaces
• Euclidean spaces
• Cauchy-Schwarz Inequality
• Minkowski’s Inequality
• Open balls
• Bounded sets
• Convergent sequences in metric spaces
• Cauchy sequences in metric spaces
• Bolzano-Weierstrass property for R^m
• Topology of metric spaces
• Open sets and closed sets
• Limit points and isolated points
• Closures and boundaries
• Subspace topology
• Limit and continuity in metric spaces
• Limit and continuity in Euclidean spaces

Scope: Sections 2.1 – 2.2, 3.1 – 3.5 and 3.5.1 of Textbook (A)

Unit-III

Completeness and Compactness

• Complete metric spaces
• Banach Contraction Principle
• Cantor Intersection Property (Nested Set Property)
• Totally bounded sets
• Compact sets
• Compact sets and closed sets
• Compact subsets of Euclidean spaces
• Finite Intersection Property
• Sequentially compact sets
• Characterizations of compact sets
• Continuity and compactness
• Uniform continuity
• Path connectedness
• Connectedness
• Continuity and connectedness

Scope: Sections 4.1 – 4.3, 5.1 – 5.3 and 6.1 – 6.3 of Textbook (A)

Unit-IV

Advanced Topics in Metric Spaces

• Countable and uncountable sets
• Applications to metric topology
• Discontinuities of monotone functions
• Introduction to separable sets
• Perfect sets
• Equivalent metrics
• Homeomorphisms
• The Cantor set

Scope: Sections 7.1, 7.2, 7.3.1, 8.1, 8.2, 9.1, 9.2 and 10.1 of Textbook (A)

Essential Textbook

(A) S. P. S. Kainth, A Comprehensive Textbook on Metric Spaces, Springer-Nature, Singapore, 2023.

Further Readings

1. T. Apostol, Mathematical Analysis (2nd Edition), Narosa, 1973.
2. R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis (3rd Edition), John Wiley & Sons, 2002.
3. N. L. Carothers, Real Analysis, Cambridge University Press, 2000.
4. R. R. Goldberg, Methods of Real Analysis, Oxford and IBH Publishing Company, 1970.
5. S. C. Malik and S. Arora, Mathematical Analysis (3rd Edition), New Age International, 2008.
6. W. Rudin, Principles of Mathematical Analysis (3rd Edition), McGraw-Hill, 1976.
7. M. O’ Searcoid, Metric Spaces, Springer, 2007.
8. S. Shirali and H. L. Vasudeva, Metric Spaces, Springer, 2006.
9. E. C. Titchmarsh, The Theory of Functions (2nd Edition), Oxford University Press, 1961.

MAT-DSC-4: Number Theory

Credits: 6 (L=5; T=1; P=0)

Total Marks: 150 (Including Internal Assessment = 30)

Time Allowed for Examination: 3 Hours

Instructions for the Candidates and Paper Setters

Candidates will be required to attempt five questions out of nine questions carrying equal marks.

Question No. 1, covering the entire syllabus, will be compulsory.

There will be two questions from each unit, and students will be required to attempt one question from each unit.

Learning Outcome

The aim of this course is to introduce students to the fundamentals of Elementary Number Theory.

The course covers topics such as prime numbers, congruences, quadratic residues, primitive roots, and arithmetic functions.

Along with theoretical concepts, emphasis will also be placed on problem-solving techniques and applications.

Unit-I

Divisibility and Congruences

• Divisibility
• Greatest Common Divisor (GCD)
• Euclidean Algorithm
• Fundamental Theorem of Arithmetic
• Congruences
• Residue classes and reduced residue classes
• Chinese Remainder Theorem
• Fermat’s Little Theorem

Unit-II

Arithmetic Functions

• Wilson’s Theorem
• Euler’s Theorem
• Applications of Euler’s Theorem in Cryptography
• Arithmetic functions:
• φ(n)
• d(n)
• σ(n)
• μ(n)
• Mobius Inversion Formula
• Greatest Integer Function

Unit-III

Quadratic Residues and Primitive Roots

• Primitive roots and indices
• Quadratic residues
• Legendre symbol
• Euler’s Criterion
• Gauss Lemma
• Quadratic Reciprocity Law
• Jacobi Symbol

Unit-IV

Diophantine Equations and Quadratic Forms

• Representation of integers as sums of two squares
• Representation of integers as sums of four squares
• Diophantine equations:
• ax + by = c
• x² + y² = z²
• x⁴ + y⁴ = z²
• Binary quadratic forms
• Equivalence of quadratic forms
• Perfect numbers
• Mersenne primes
• Fermat numbers

Essential Textbooks

(A) D. M. Burton, Elementary Number Theory, 6th Edition, Tata McGraw Hill, 2007.

(B) I. Niven, H. S. Zuckerman and H. L. Montgomery, An Introduction to the Theory of Numbers, 5th Edition, John Wiley and Sons, 2004.

Further Readings

1. H. Davenport, The Higher Arithmetic, 7th Edition, Cambridge University Press, 1999.
2. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th Edition, Oxford University Press, 2008.