Rubik's Cube Example

Factorial Example

Paper Folding Example

Motivation for Counting

Motivation for Catalan numbers

Counting in Computer Science

Rule of Sum and Rule of Product

Problems on Rule of Sum and Rule of Product

Factorial Explained

Proof of n! - Part 1

Proof of n! - Part 2

Permutations - Part 1

Astronomical Numbers

Problems on Permutations

Permutations - Part 4

Permutations - Part 3

Permutations - Part 2

Combinations - Part 2

Combinations - Part 1

Combinations - Part 3

Combinations - Part 4

Problems on Combinations

Difference between Permuations and Combinations

Combination with Repetition - Part 1

Combination with Repetition - Part 2

Combination with Repetition - Problems

Problems on Binomial theorem

Multinomial theorem

Chapter Summary

Examples of Catalan numbers

Catalan Numbers - Part 4

Catalan Numbers - Part 2

Catalan Numbers - Part 1

Fun facts on Pascal's Triangle

Catalan Numbers - Part 3

Properties of Binomial theorem

Pascal's Triangle

Binomial theorem

Applications of Binomial theorem

Introduction to Set Theory

Example, definiton and notation

Sets - Problems Part 1

Subsets - Part 1

Subsets - Part 2

Subsets - Part 3

Union and intersections of sets

Union and intersections of sets - Part 1

Union and intersections of sets - Part 2

Summary

Symmetric difference

History

Set difference - Part 2

Set difference - Part 1

De Morgan's Laws - Part 4

De Morgan's Laws - Part 3

A proof technique

De Morgan's Laws - Part 2

De Morgan's Laws - Part 1

Complement of a set

Power set - Problems

Connection betwenn Binomial Theorem and Power Sets

Power set - Part 3

Power set - Part 2

Power Set - Part 1

Cardinality of Union of three sets

Cardinality of Union of sets - Part 2

Cardinality of Union of two sets - Part 1

Union and intersections of sets - Part 3

Truthtable for OR operator

Introduction to OR operator

Motivation for OR operator

Examples for Negation

Negation - Truthtable

Negation - Explanation

Examples and Non-examples of Statements

Introduction to Statements

Introduction to Negation

Motivational example

OR operator for 3 Variables

Truthtable for AND operator

Primitive and Compound statements - Part 2

Primitive and Compound statements - Part 1

AND operator for 3 Variables

Problems involoving NOT, OR and AND operators

Examples and Non-examples of Implication - Part 1

Introduction to implication

Examples and Non-examples of Implication - Part 2

Explanation of Implication

Introduction to Double Implication

Logical Equivalence - Part 3

Logical Equivalence - Part 2

Logical Equivalence - Part 1

SAT Problem - Part 2

SAT Problem - Part 1

Tautology, Contradiction - Part 3

Tautology, Contradiction - Part 2

Tautology, Contradiction - Part 1

Double negation - Part 2

Double negation - Part 1

Motivation for laws of logic

XOR operator - Part 3

XOR operator - Part 2

XOR operator - Part 1

Converse, Inverse and Contrapositive

Problems

Explanation of Double Implication

Laws of Logic

Logical Equivalence - Part 4

De Morgan's Law - Part 2

De Morgan's Law - Part 1

Conclusion

Rules of Inferences - Part 7

Rules of Inferences - Part 6

Rules of Inferences - Part 5

Rules of Inferences - Part 4

Rules of Inferences - Part 3

Rules of Inferences - Part 2

Rules of Inferences - Part 1

Number of relations - Part 1

Examples of Relations

Revisiting Representations of a Relation

Set Representation of a Relation

Cartesian Product

Relation - An Example

Matrix Representation of a Relation

Various sets

Graphical Representation of a Relation

Introduction to Relation

Number of symmetric relations - Part 1

Symmetric Relation - Examples and non examples

Parallel lines revisited

Symmetric Relation - Matrix representation

Symmetric Relation - Introduction

Number of Reflexive relations

Reflexive relation - Matrix representation

Example of a Reflexive relation

Reflexive relation - Introduction

Number of relations - Part 2

Condition for relation to be reflexive

Number of Antisymmetric relations

Antisymmetric - Matrix representation

Antisymmetric - Graphical representation

Examples of Transitive and Antisymmetric Relation

Antisymmetric relation

Transitive relation - Examples and non examples

Pattern

Number of symmetric relations - Part 2

Examples of Reflexive and Symmetric Relations

Partition - Part 5.

Partition - Part 5

Partition - Part 4

Partition - Part 3

Partition - Part 2

Partition - Part 1

Equivalence relation - Example 4

Equivalence relation

Condition for relation to be antisymmetric

Condition for relation to be symmetric.

Condition for relation to be symmetric

Condition for relation to be reflexive.

Condition for relation to be reflexive..

Few notations

Introduction to Onto Function - Part 1

Cardinality condition in One-One function - Part 1

Cardinality condition in One-One function - Part 2

Examples and Non- examples of One-One function

One-One Function - Example 3

Proving a Function is One-One

One-One Function - Example 2

One-One Function - Example 1

Introduction to One-One Function

Relations vs Functions - Part 2

Relations vs Functions - Part 1

Defintion of a function - Part 3

Defintion of a function - Part 2

Defintion of a function - Part 1

Introduction to functions

Example - 4 Explanation

Motivational Example - 3

Commonality in examples

Motivational Example - 2

Motivational Example - 1

Number of functions

Counting number of functions

Cardinality condition in Bijection - Part 2

Cardinality condition in Bijection - Part 1

Examples of Bijection

Introduction to Bijection

Cardinality condition in Onto function - Part 2

Cardinality condition in Onto function - Part 1

Examples of Onto Function

Definition of Onto Function

Introduction to Onto Function - Part 2

Application of inverse functions - Part 1

Examples of Inverse functions

Inverse functions

Motivation for Inverse functions

Example of Composition of functions - Part 2

Example of Composition of functions - Part 1

Why study Composition of functions

Definition of Composition of functions

Motivation for Composition of functions - Part 2

Motivation for Composition of functions - Part 1

Counting number of functions.

Number of Bijections

Number of Onto functions

Number of One-One functions - Part 3

Number of One-One functions - Part 2

Number of One-One functions - Part 1

Checker board and triominoes - Solution

Checker board and Triomioes - A puzzle

Binomial Coeffecients - Proof by induction

Mathematical Induction - Example 10 solution

Mathematical Induction - Example 9

MI - Inequality 2 solution

MI - Inequality 2

MI - Problem on satisfying inequalities (solutions)

MI - Problem on satisfying inequalities

MI - To prove divisibility

MI - Inequality 1 (solution)

MI - Inequality 1

MI - To prove divisibility (solution)

MI - Sum of powers of 2

MI - Sum of odd numbers

Mathematical Induction - The formal way

Mathematical Induction - Its essence

Mathematical induction - An illustration

Three stories

Three stories - Connecting the dots

10 points on an equilateral triangle

Set of n integgers

Group of n people

Motivation for Pegionhole Principle

A false proof - Solution

Mathematical induction - An important note

Mathematical Induction - A false proof

Pegionhole Principle - A result

Consecutive integers

Consecutive integers solution

Matching initials

Numbers adding to 9

Matching initials - Solution

Numbers adding to 9 - Solution

Deck of cards

Number of errors - Solution

Number of errors

Deck of cards - Solution

Puzzle - Challenge for you

Subgraph

Relation between walk and path - An induction proof

Relation between walk and path

Example of cycle and circuit

Cycle and circuit

Examples of walk, trail and path

Definitions revisited

Path and closed path

Walk

Regular graph and irregular graph

Trail

Havel Hakimi theorem - Part 5

Havel Hakimi theorem - Part 4

Havel Hakimi theorem - Part 3

Havel Hakimi theorem - Part 2

Havel Hakimi theorem - Part 1

Problems based on Hand shaking lemma

Hand shaking lemma - Corollary

Relation between number of edges and degrees - Proof

Relation between number of edges and degrees

Defintion of a Graph

Coloring the India map

Degree and degree sequence

Connectedness through Connecting people

Traversing the bridges

Three utilities problem

Friendship - an interesting property

Introduction to Python - Installation

NetworkX - Need of the hour

Illustration of cut vertices and cut edges

Cut edge

Cut vertex

Connecting connectedness and path - An illustration

Connecting connectedness and path

Edge condition for connectivity

Property of a cycle

Connected and Disconnected graphs

Introduction to Tree

Spanning and induced subgraph - A result

Spanning and induced subgraph

Introduction to Python - Basics

Introduction to NetworkX

Story so far - Using NetworkX

Graph representations - Introduction

Illustration of Directed, weighted and multi graphs

Directed, weighted and multi graphs

Adjacency matrix representation

Isomorphism - Introduction

Incidence matrix representation

Isomorphic graphs - An illustration

Isomorphic graphs - A challenge

Non - isomorphic graphs

Isomorphism - A question

Complement of a Graph - Introduction

Complement of a Graph - Illiustration

Self complement

Complement of a disconnected graph is connected

Which is more? Connected graphs or disconnected graphs?

Complement of a disconnected graph is connected - Solution

Definition of Eulerian Graph

Bipartite graphs - Converse part of the puzzle

Bipartite graphs - A puzzle

Bipartite graphs

Planar graphs - Inequality 1

3 Utilities problem - Revisited

Litmus test for planarity

Famous non-planar graphs

Proof of V - E + R = 2

V - E + R = 2; Use induction

Illustration of V - E + R =2

V - E + R = 2

Examples of Planar graphs

Definition of a Planar graph

Constructing non intersecting roads

Importance of Hamiltonian graphs in Computer science

Eulerian and Hamiltonian Are they related

Dirac's Theorem v/s Ore's Theorem

Ore's Theorem

Dirac's theorem - A note

Dirac's Theorem

A result on Path

A result on connectedness

Hamiltonian graph - A result

Examples of Hamiltonian graphs

Defintion of Hamiltonian graphs

Can you traverse all location?

Why the name Eulerian

A condition for Eulerian trail

Proof for even degree implies graph is eulerian

Why even degree?

Litmus test for an Eulerian graph

Non- example of Eulerian graph

Illustration of eulerian graph

Recalling the India map problem - Solution

Recalling the India map problem

Examples on Proper coloring

Chromatic number of a graph

Prisoners example and Proper coloring

NetworkX - Digraphs

NetworkX - Adjacency matrix

NetworkX- Random graphs

NetworkX - Subgarph

NetworkX - Isomorphic graphs: A game to play

NetworkX - Isomorphic graphs Part 1

NetworkX - Isomorphic graphs Part 2

NetworkX - Graph complement

Counting in a creative way

NetworkX - Coloring

NetworkX - Bipaprtite graphs

NetworkX - Eulerian graphs

Picking five balls - Solution

Example 2 - Picking five balls

Picking five balls - Another version

Words and the polynomial - Explained

Words and the polynomial

Example 1 - Fun with words

Defintion of Generating function

Generating function examples - Part 2

Generating function examples - Part 1

Picking 7 balls - The naive way

Binomial expansion - Explained

Binomial expansion - A generating function

Generating function examples - Part 3

Generating functions - Problem 1

Picking 7 balls - The creative way

Generating functions - Problem 2

Generating functions - Problem 3

Why Generating function?

Introduction to Advanced Counting

Example 1 : Dogs and Cats

Inclusion-Exclusion Formula

Proof of Inclusion - Exlusion formula

Example 2 : Integer solutions of an equation

Example 3 : Words not containing some strings

Example 4 : Arranging 3 x's, 3 y's and 3 z's

Example 5 : Non-multiples of 2 or 3

Example 6 : Integers not divisible by 5, 7 or 11

A tip in solving problems

Example 7 : A dog nor a cat

Example 8 : Brownies, Muffins and Cookies

Example 10 : Integer solutions of an equation

Example 11 : Seating Arrangement - Part 1

Example 11 : Seating Arrangement - Part 2

Example 12 : Integer solutions of an equation

Number of Onto Functions.

Formula for Number of Onto Functions

Example 13 : Onto Functions

Example 14 : No one in their own house

Derangements

Derangements of 4 numbers

Example 15 : Bottles and caps

Example 16 : Self grading

Example 17 : Even integers and their places

Example 18 : Finding total number of items

Example 19 : Devising a secret code

Placing rooks on the chessboard

Rook Polynomial

Rook Polynomial.

Motivation for recurrence relation

Getting started with recurrence relations

Recurrence relation for Merge sort

What is a recurrence relation?

Compound Interest as a recurrence relation

Examples of recurrence relations

Example - Number of ways of climbing steps

Number of ways of climbing steps: Recurrence relation

Example - Rabbits on an island

Example - n-bit string

Example - n-bit string without consecutive zero

Solving Linear Recurrence Relations - A theorem

A note on the proof

Soving recurrence relation - Example 1

Soving recurrence relation - Example 2

Fibonacci Sequence

Introduction to Fibonacci sequence

Solution of Fibbonacci sequence

A basic introduction to 'complexity'

Intuition for 'complexity'

Visualizing complexity order as a graph

Tower of Hanoi

Reccurence relation of Tower of Hanoi

Solution for the recurrence relation of Tower of Hanoi

A searching technique

Recurrence relation for Binary search

Solution for the recurrence relation of Binary search

Example: Door knocks example

Example: Door knocks example solution

Door knock example and Merge sort

Introduction to Merge sort - 1

Correspondence between partition and generating functions: In general

Correspondence between partition and generating functions

Chromatic polynomial of cycle on 4 vertices - Part 2

Chromatic polynomial of cycle on 4 vertices - Part 1

Chromatic polynomial of complete graphs

Introduction to Chromatic polynomial

Intoduction to advanced topics

Groups: Examples and non-examples

Formal definition of a Group

Uniqueness of the identity element

Introduction to Group Theory

Recurrrence relation: The theorem and its proof

Motivation for exponential generating function

Example: Picking 4 letters from the word 'INDIAN'

Why 'partitions' to 'polynomial'?

Distinct partitions equals odd partitions: Proof

Distinct partitions equals odd partitions: Observation

Odd partitions and generating functions

Distinct partitions and generating functions

Distinct partitions and odd partitions

Conclusion.

Summary.

Lagrange's theorem

Subgroup: Defintion and examples

Groups: Special Examples Part 2

Groups: Special Examples Part 1