Introduction to Algebraic Structures WP1B7JMCGE

URI	`m2://res/WP1B7JMCGE`
Kind	course
Creators	Mahesh Kakde (instructor), R. Venkatesh (instructor)
Date	2026-06-17
Scope	in
Ids	course 04dese585:111108001

Expressions

e1: edition

Language	en
Date	2026-06-17

m1: page, IISc Bangalore (record). nptel 111108001
m2: page, 2026-06-17 (record). nptel 111108001

Contents

part 1: Week 1 Introduction to set theory and functions
session 1: Introduction to mathematics and set theory
session 2: Set identities, Cartesian product, relations and functions
session 3: Logic and construction of new sets
session 4: Functions
session 5: Properties of functions
part 2: Week 2 Further set theory and the use of functions
session 6: Composing functions and Cantor's theorem
session 7: Don Zaiger's proof: Primes of the form 4k+1 can written as sum of two squares.
session 8: Axiom of choice and Paritition of a set.
session 9: Correspondence between partitions of a set and equivalence relations on it.
session 10: Partially ordered sets, well-ordering principle and Zorn's lemma.
part 3: Week 3: Natural numbers, order properties, finite sets, countably infinite sets and uncountable sets")
session 11: Defining natural numbers: Peano axioms.
session 12: Order properties of natural numbers and notion of cardinality of a set.
session 13: Cardinality of a finite set.
session 14: Countable sets
session 15: Uncountable sets
part 4: Week 4 Foundations of Number Systems and Counting Techniques
session 16: Schroder-Bernstein theorem
session 17: Integers and Rational Numbers
session 18: The Construction of Rational Numbers and Counting Principles
session 19: Counting in Two Ways
session 20: Binomial Coefficients and Properties
part 5: Week 5 Pigeonhole Principle, Formal Series, Linear Recurrence Relations, Inclusion–Exclusion Principle
session 21: The Pigeonhole Principle
session 22: Formal Series
session 23: Formal Series and Applications
session 24: Solving Linear Recurrences
session 25: Principle of Inclusion and Exclusion
part 6: Week 6 Partially Ordered Sets, Möbius Function, Graph Theory
session 26: Möbius Function on Finite Posets
session 27: Möbius Inversion Formula
session 28: Möbius Inversion Formula and Applications
session 29: Introduction to Graph Theory, Graphs and Special Graph Families
session 30: Graph Representations and Basic Graph Operations
part 7: Week 7 Graph Structures and Properties
session 31: Handshake Lemma and Graph Connectivity
session 32: Cycles, Bipartite Graphs, and Trees
session 33: Trees and Their Properties
session 34: Eulerian and Hamiltonian Graphs
session 35: Matchings and Graph Coloring
part 8: Week 8 Introduction to number theory
session 36: Number theory and divisibility
session 37: Greatest Common Divisor - I
session 38: Greatest Common Divisor - II
session 39: LCM, primes and prime factorisation
session 40: Consequences of prime factorisation and infinitude of primes
part 9: Week 9 Modular arithmetic
session 41: Euler's proof of infinitude of primes
session 42: Congruencies and modular arithmetic
session 43: Diophantine equations and residue classes
session 44: Solving Diophantine equations in one variable and Chinese remainder theorem
session 45: Solving Diophantine equations using CRT
part 10: Week 10 Congruencies and Mobius inversion
session 46: Proof of Hensel's lemma
session 47: Introduction to Euler's totient function
session 48: More on totient function
session 49: Mobius inversion formula
session 50: Euler's theorem
part 11: Week 11 Ring of reside classes and its group of units, polynomial ring over a field, primitive roots and linear congruences.
session 51: Ring structure on set of residue classes
session 52: Polynomial ring over a field, division algorithm and consequences.
session 53: Wilson's theorem, Lagrange's theorem and introduction to cyclic groups
session 54: Classification of cyclic unit groups modulo m and proof.
session 55: Primitive roots modulo m and application in solving linear congruences
part 12: Week 12 Quadratic residue, Legendre symbol and primes in AP.
session 56: More on solving congruence equations and introduction to quadratic residue and Legendre symbol
session 57: Computing Legendre symbol: Euler's criterion and Law of quadratic reciprocity
session 58: Dirichlet's theorem on primes in AP: Proof of special cases

Catalogue records

nptel: 111108001 (manifestation)
nptel_video: 111108001 (manifestation)