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.
Ordered Sets & Lattices
Understand partially ordered sets, Hasse diagrams, and lattice theory in mathematical structures.
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 Dijkstraโs 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- Fleuryโs algorithm
- Eulerian graphs- Chinese-Postman-Problem Hamilton
Matching
Learn about matching in graphs, approximation algorithms, and optimization techniques.
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.
Quick Links
Access last-minute notes and quizzes to reinforce your learning in discrete mathematics.
