Discrete Math Course

Lesson 1 • Mathematical Notation and Symbols

Covers standard symbols, quantifiers, and summation notation used in discrete math. Builds reading and writing fluency for formal mathematical statements.

Lesson 2 • Introduction to Logic and Propositions

Defines propositions, logical connectives, and truth values. Connects logical reasoning to proof writing and algorithm analysis.

Lesson 3 • Sets and Set Operations

Introduces sets, membership, and operations such as union, intersection, and complement. Provides the foundational language for all subsequent topics.

Lesson 1 • Prime Numbers and Factorization

Examines prime distribution, primality testing, and unique factorization. Supports understanding of RSA encryption and number-theoretic algorithms.

Lesson 2 • Modular Arithmetic

Introduces congruences, modular operations, and residue classes. Provides the arithmetic foundation for cryptographic and hashing applications.

Lesson 3 • Divisibility and Integer Properties

Defines divisibility, factors, and multiples for integers. Establishes properties used in GCD computation and modular arithmetic.

Lesson 1 • Normal Forms and Simplification

Covers conjunctive and disjunctive normal forms and Boolean simplification. Connects logic simplification to circuit design and automated reasoning.

Lesson 2 • Propositional Logic and Inference

Formalizes propositional logic with inference rules and logical laws. Builds the ability to derive conclusions from given premises systematically.

Lesson 3 • Logic in Algorithms and Programs

Applies logical reasoning to program correctness, loop invariants, and specifications. Bridges formal logic to software verification and algorithm analysis.

Lesson 1 • Recursive Definitions and Structures

Defines sequences, sets, and functions recursively. Connects recursive definitions to data structures such as lists and trees.

Lesson 2 • Strong Induction and Well-Ordering

Extends induction to strong form and connects it to the well-ordering principle. Enables proofs where multiple prior cases are needed simultaneously.

Lesson 3 • Solving Recurrence Relations

Covers methods for finding closed-form solutions to linear recurrences. Supports algorithm complexity analysis through recurrence solving.

Lesson 1 • Pigeonhole Principle and Applications

States the pigeonhole principle and applies it to existence proofs. Demonstrates how counting arguments guarantee structural properties in discrete systems.

Lesson 2 • Generating Functions Introduction

Introduces ordinary generating functions as a counting tool. Connects polynomial algebra to combinatorial enumeration and recurrence solving.

Lesson 3 • Permutations and Combinations

Distinguishes ordered and unordered selections and derives factorial-based formulas. Applies these to scheduling, arrangement, and selection problems.

Lesson 4 • Binomial Theorem and Identities

Proves the binomial theorem and derives combinatorial identities. Connects algebraic expansion to counting arguments and Pascal's triangle.

Lesson 1 • Equivalence Relations and Partitions

Connects equivalence relations to partitions of a set into disjoint classes. Applies equivalence classes to modular arithmetic and type systems.

Lesson 2 • Functions: Types and Properties

Classifies functions as injective, surjective, and bijective and analyzes inverses. Supports understanding of data encoding, hashing, and cryptographic mappings.

Lesson 3 • Binary Relations and Properties

Defines binary relations and their properties: reflexivity, symmetry, and transitivity. Provides the framework for equivalence and order relations.

Lesson 1 • Trees and Spanning Trees

Defines trees, rooted trees, and spanning trees and proves key properties. Connects tree structures to hierarchical data and minimum spanning tree algorithms.

Lesson 2 • Graph Properties and Special Graphs

Examines degree sequences, connectivity, and special graph families. Connects structural properties to algorithm design and network analysis.

Lesson 3 • Euler and Hamiltonian Paths

States conditions for Euler and Hamiltonian paths and circuits. Applies these concepts to routing, scheduling, and network design problems.

Lesson 4 • Graph Definitions and Representations

Defines vertices, edges, directed and undirected graphs, and common representations. Establishes vocabulary and data structures for all graph algorithms.

Lesson 1 • Network Flow and Max-Flow Min-Cut

Introduces flow networks, feasible flows, and the max-flow min-cut theorem. Connects network flow to transportation, routing, and resource allocation.

Lesson 2 • Matching and Bipartite Graphs

Covers perfect matchings, Hall's theorem, and augmenting path algorithms. Applies matching theory to assignment and scheduling optimization.

Lesson 3 • Shortest Path Algorithms

Covers Dijkstra's and Bellman-Ford algorithms for weighted graph shortest paths. Applies these to routing protocols and geographic navigation systems.

Lesson 4 • Planar Graphs and Graph Isomorphism

Proves Euler's formula for planar graphs and introduces isomorphism testing. Connects planarity to circuit layout and graph database queries.