Syllabus (MATHEMATICS)

Course Type: MAJ-2

Semester: 2

Course Code: BMTMMAJ02T

Course Title: Algebra-I & Real Analysis-I

(L-P-Tu): 5-0-1

Credit: 6

Practical/Theory: Theory

Course Objective: TO BE DONE

Learning Outcome: The whole course will have the following outcomes: CO1: Objective of this course is the revision of their previous lessons of relation, equivalence relation, mapping, inverse mapping and their properties. CO2: Group theory is one of the building blocks of

Syllabus:

Unit -1: Algebra-I [Credit: 2, 30L]

• Relation: Equivalence relation, equivalence classes and partition, partial ordered relation. Hesse’s diagram, Lattices as partially ordered set, definition of lattice in terms of meet and join, equivalence of two definitions, linear order relation.
• Mappings: Injective, surjective, one-to-one correspondence, composition of two mappings. Inversion of mappings. Extension and restriction of a mappings.

• Principles of Mathematical Induction, Well-ordering Principle, Division Algorithm, Greatest common divisor, Primes and composite numbers, Fundamental theorem of arithmetic, relatively prime numbers, Euclid’s algorithm, least common multiple.
• Congruences: Properties and algebra of congruences, power of congruence, Fermat’s congruence, Fermat’s theorem, Wilson’s theorem, Euler – Fermat’s theorem, Chinese remainder theorem, Number of divisors of a number and their sum, least number with given number of divisors.
• Eulers φ function-φ(n). Mobius μ-function, relation between φ function and μ function. Diophantine equations of the form ax+by = c, a, b, c are integers.
• Group: Groupoid, semigroup, monoid and quasigroup (definitions and examples). Group, uniqueness of identity and inverse element, law of cancellation, order of a group and order of an element, abelian group, elementary properties of group.
• Definitions of rings and fields with examples.

Unit -2: Real Analysis-I [Credit: 4, 60L]

• Review of Algebraic and Order Properties of R, ε-neighbourhood of a point in R. Idea of countable sets and countability of R, uncountable sets and uncountability of R. Boundedness. Suprema and Infima. Completeness Property of R and its equivalent properties. The Archimedean Property, Density of Rational (and Irrational) numbers in R.

• Point Set Theory in R: Intervals. Interior points of a set, open sets. Limit points of a set, isolated points, closed set, derived set, Illustrations of Bolzano-Weierstrass theorem for sets.

• Sequences: Real sequence, Bounded sequence, Limit of a sequence, Convergent sequence, liminf, limsup. Limit Theorems. Monotone Sequences, Monotone Convergence Theorem. Sandwich Rule, Nested interval theorem, Cauchy’s first and second limit theorem. Subsequences, Divergence Criteria. Monotone Subsequence Theorem (statement only), Bolzano Weierstrass Theorem for Sequences. Cauchy sequence, Cauchy’s Convergence Criterion.

• Infinite series: Convergence and divergence of infinite series, Cauchy Criterion, Tests for convergence: Comparison test, Limit Comparison test, Ratio test, Cauchy’s nth root test, Raabe’s test (proof not required), Gauss’s test (proof not required), Cauchy’s condensation test (proof not required), Integral test (proof not required). Alternating series, Leibnitz test (proof not required). Absolute and Conditional convergence. Re-arrangement of terms (concepts and elementary examples only).

• Limits: Concepts of limits of a function (ε-δ approach), sequential criterion for limits, divergence criteria. Limit theorems, one sided limits. Infinite limits and limits at infinity.

• Continuity: Continuous functions, sequential criterion for continuity and discontinuity. Algebra of continuous functions. Continuous functions on an interval, intermediate value theorem, location of roots theorem, preservation of intervals theorem. Uniform continuity, non-uniform continuity criteria, uniform continuity theorem.

• Differentiability: Differentiability of a function at a point and in an interval, Caratheodory’s theorem, algebra of differentiable functions. Relative extrema, interior extremum theorem. Rolle’s theorem. Mean value theorem. Intermediate value property. Darboux’s theorem (statement only). Applications of mean value theorem to inequalities and approximation of polynomials. Cauchy’s mean value theorem. Taylor’s theorem with Lagrange’s form of remainder, Taylor’s theorem with Cauchy’s form of remainder, application of Taylor’s theorem to convex functions, relative extrema. Taylor’s series and Maclaurin’s series expansions of exponential and trigonometric functions. Application of Taylor’s theorem to inequalities.

Reading References:

1. Sen, Ghosh, Mukhopadhaya, Maity; Topics in Abstract Algebra; Universities Press.
2. S. K. Mapa, Higher Algebra (Abstract and Linear), Levant.

1. John B. Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002
2. Joseph A. Gallian; Contemporary Abstract Algebra; 9th Ed..; Narosa Pub. H.; New Delhi, 1999.
3. D.S. Malik, John M. Mordeson and M.K. Sen; Fundamentals of abstract algebra; McGraw-Hill.
4. T.M. Apostol; Introduction to Analytic Number Theory; Springer.
5. I. Niven, H.S. Zuckerman, H.L. Montgomery; An Introduction to Theory of Numbers; 5th Ed.; John Wiley & Sons, Inc.
6. A.K. Chowdhury; Introduction to Number Theory; 2^nd Ed.; NCBA.
7. R.G. Bartle and D. R. Sherbert; Introduction to Real Analysis; 3rd Ed.; John Wiley and Sons (Asia) Pvt. Ltd.; Singapore; 2002.
8. S.K. Berberian; A First Course in Real Analysis; Springer-Verlag; New York; 1994.
9. W. Rudin; Principles of Mathematical Analysis; Tata McGraw-Hill.
10. S.K. Mapa; Introduction to Real analysis; Levant.
11. S.C. Malik & Savita Arora; Mathematical Analysis; New Age International (P) Limited.
12. S.R. Ghorpade and B.V. Limaye; A Course in Calculus and Real Analysis: Springer: 2006.
13. Tom M. Apostol; Mathematical Analysis; Narosa Publishing House.
14. Charles G. Denlinger; Elements of Real Analysis; Jones & Bartlett (Student Edition); 2011.
15. Richard R. Goldberg; Methods of Real Analysis; 2nd Ed.; John Wiley and Sons, Inc.