Discrete Mathematics Tutorial

Discrete Mathematics is a branch of mathematics that is concerned with "discrete" mathematical structures instead of "continuous" ones. Discrete mathematical structures include objects with distinct values, like graphs, integers, logic-based statements, etc.

To explore these concepts in detail, we begin with the following core topics.

Mathematical Logic

Learn propositional and predicate logic, equivalences, proofs, and rules of inference for logical reasoning.

• Introduction to Propositional Logic
• Applications of Propositional Logic
• Propositional vs Predicate Logic
• Propositional Equivalences
• Normal and Principal Forms
• Predicates and Quantifiers
• Nested Quantifiers Theorem
• Rules of Inference
• Introduction to Proofs

Sets and Relations

Understand set theory, operations, relations, functions, and equivalence relations with real-world applications.

• Set Theory
• Types of Sets
• Set Operations
• Rough Set Theory
• Functions
• Sequence and Summations
• Representations of Matrices and Graphs in Relations
• Types of Relation
• Closure of Relation and Equivalence Relations

Mathematical Inductions

Explore counting techniques, permutations, combinations, generating functions, and probability concepts.

• Mathematical Induction
• Basics of Counting
• Pascal's Identity
• Pigeonhole Principle
• Permutations and Combinations
• Generalized Permutations and Combinations
• Generating Functions
• Inclusion-Exclusion Principle
• Discrete Probability Theory

Boolean Algebra

Study Boolean functions, algebraic theorems, properties, and methods for minimizing Boolean expressions.

• Boolean Functions
• Boolean Algebraic Theorem
• Properties of Boolean Algebra
• Number of Boolean Functions
• Minimization of Boolean Functions

Optimization

Learn linear programming, the simplex algorithm, and PERT for solving optimization problems.

• Linear Programming 
• Graphical Solution For Linear Programming
• Simplex Algorithm
• PERT 

Ordered Sets & Lattices

Understand partially ordered sets, Hasse diagrams, and lattice theory in mathematical structures.

• Partially Ordered Sets
• Elements of POSET
• Hasse Diagrams
• Lattices

Probability Theory

Dive into probability axioms, conditional probability, and common distributions like Poisson, normal, and exponential.

• Basic Concepts of Probability
• Probability Axioms
• Properties of Probability
• Conditional Probability
• Bayes' Theorem
• Uniform Distribution
• Exponential Distribution
• Normal Distribution
• Poisson Distribution

Graph Theory

Explore graph types, algorithms (like Dijkstra’s and DFS), and applications in problem-solving and optimization.

• Introduction to Graph
• Basic Terminology of a Graph
• Types of a Graph
• Walks, Trails, Paths, and Circuits
• Graph Distance components
• Cut-Vertices and Cut-Edges
• Bridge in Graph
• Independent sets
• Shortest Path Algorithms [Dijkstra's Algorithm]
• Application of Graph Theory
• Graph Traversals[DFS]
• Graph Traversals[BFS]
• Characterizations of Trees
• Prim's Minimum Spanning Tree 
• Kruskal's Minimum Spanning Tree
• Huffman Codes 
• Tree Traversals
• Traveling Salesman Problem

Special Graph

Study Eulerian and Hamiltonian graphs, including algorithms and problems like the Traveling Salesman Problem.

• Bipartite Graphs 
• Independent Sets and Covering
• Eulerian graphs
• Eulerian graphs- Fleury’s algorithm
• Eulerian graphs- Chinese-Postman-Problem Hamilton

Matching

Learn about matching in graphs, approximation algorithms, and optimization techniques.

• Matching- Basics, Perfect, Bipartite
• Approximation Algorithms

Vertex Colorings

Explore graph colorings, chromatic numbers, and algorithms like Greedy Coloring for graph theory.

• Chromatic Numbers, Greedy Coloring Algorithm
• Edge Coloring
• Vizing Theorem
• Planar Graph- Basics, Planarity Testing
• Directed Graphs- Degree Centrality
• Directed Graphs- Weak Connectivity
• Directed Graphs- Strong Components 
• Directed Graphs- Eulerian, Hamiltonian
• Directed Graphs- Tarjan's Algorithm
• Handshaking in Graph Theorem

Group Theory

Understand the basics of groups, subgroups, isomorphisms, and structures like rings and fields.

• Groups, Subgroups, Semi-Groups
• Isomorphism, Homomorphism
• Automorphism
• Rings, Integral domains, Fields

Quick Links

Access last-minute notes and quizzes to reinforce your learning in discrete mathematics.

• Last-Minute Notes (LMNs)
• Quizzes on Discrete Mathematics