Learn · NPTEL

Course11110800158 lectures

Introduction to Algebraic Structures

Prof. Mahesh Kakde and Prof. R. Venkatesh · IISc Bangalore

Start with lecture 1 At NPTEL

About

This course aims to offer an accessible introduction to algebraic structures to undergraduate students, exploring foundational topics such as set theory, combinatorics, graph theory, number theory, and group theory. It presents fundamental counting techniques and applies them to solve introductory counting problems, including those related to graphs. Gradually, the course transitions into basic number theory and introduces the concept of groups, using examples derived from combinatorics and number theory to provide a deeper understanding of these mathematical structures. This course does not require any prior knowledge of mathematics.

Intended audience: UG level course

Lectures

Week 1 Introduction to set theory and functions

  1. Introduction to mathematics and set theory YouTube
  2. Set identities, Cartesian product, relations and functions YouTube
  3. Logic and construction of new sets YouTube
  4. Functions YouTube
  5. Properties of functions YouTube

Week 2 Further set theory and the use of functions

  1. Composing functions and Cantor's theorem YouTube
  2. Don Zaiger's proof: Primes of the form 4k+1 can written as sum of two squares. YouTube
  3. Axiom of choice and Paritition of a set. YouTube
  4. Correspondence between partitions of a set and equivalence relations on it. YouTube
  5. Partially ordered sets, well-ordering principle and Zorn's lemma. YouTube

Week 3: Natural numbers, order properties, finite sets, countably infinite sets and uncountable sets")

  1. Defining natural numbers: Peano axioms. YouTube
  2. Order properties of natural numbers and notion of cardinality of a set. YouTube
  3. Cardinality of a finite set. YouTube
  4. Countable sets YouTube
  5. Uncountable sets YouTube

Week 4 Foundations of Number Systems and Counting Techniques

  1. Schroder-Bernstein theorem YouTube
  2. Integers and Rational Numbers YouTube
  3. The Construction of Rational Numbers and Counting Principles YouTube
  4. Counting in Two Ways YouTube
  5. Binomial Coefficients and Properties YouTube

Week 5 Pigeonhole Principle, Formal Series, Linear Recurrence Relations, Inclusion–Exclusion Principle

  1. The Pigeonhole Principle YouTube
  2. Formal Series YouTube
  3. Formal Series and Applications YouTube
  4. Solving Linear Recurrences YouTube
  5. Principle of Inclusion and Exclusion YouTube

Week 6 Partially Ordered Sets, Möbius Function, Graph Theory

  1. Möbius Function on Finite Posets YouTube
  2. Möbius Inversion Formula YouTube
  3. Möbius Inversion Formula and Applications YouTube
  4. Introduction to Graph Theory, Graphs and Special Graph Families YouTube
  5. Graph Representations and Basic Graph Operations YouTube

Week 7 Graph Structures and Properties

  1. Handshake Lemma and Graph Connectivity YouTube
  2. Cycles, Bipartite Graphs, and Trees YouTube
  3. Trees and Their Properties YouTube
  4. Eulerian and Hamiltonian Graphs YouTube
  5. Matchings and Graph Coloring YouTube

Week 8 Introduction to number theory

  1. Number theory and divisibility YouTube
  2. Greatest Common Divisor - I YouTube
  3. Greatest Common Divisor - II YouTube
  4. LCM, primes and prime factorisation YouTube
  5. Consequences of prime factorisation and infinitude of primes YouTube

Week 9 Modular arithmetic

  1. Euler's proof of infinitude of primes YouTube
  2. Congruencies and modular arithmetic YouTube
  3. Diophantine equations and residue classes YouTube
  4. Solving Diophantine equations in one variable and Chinese remainder theorem YouTube
  5. Solving Diophantine equations using CRT YouTube

Week 10 Congruencies and Mobius inversion

  1. Proof of Hensel's lemma YouTube
  2. Introduction to Euler's totient function YouTube
  3. More on totient function YouTube
  4. Mobius inversion formula YouTube
  5. Euler's theorem YouTube

Week 11 Ring of reside classes and its group of units, polynomial ring over a field, primitive roots and linear…

  1. Ring structure on set of residue classes YouTube
  2. Polynomial ring over a field, division algorithm and consequences. YouTube
  3. Wilson's theorem, Lagrange's theorem and introduction to cyclic groups YouTube
  4. Classification of cyclic unit groups modulo m and proof. YouTube
  5. Primitive roots modulo m and application in solving linear congruences YouTube

Week 12 Quadratic residue, Legendre symbol and primes in AP.

  1. More on solving congruence equations and introduction to quadratic residue and Legendre symbol YouTube
  2. Computing Legendre symbol: Euler's criterion and Law of quadratic reciprocity YouTube
  3. Dirichlet's theorem on primes in AP: Proof of special cases YouTube

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IISc Bangalore
Department
Mathematics
Taught by
Prof. Mahesh Kakde, Prof. R. Venkatesh
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
Who can watch
NPTEL gives no rating. Invisible Network shows it to Tweens profiles and older, judged from what the source says about it. How courses are placed