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Discrete Math

Build a rigorous foundation for discrete structures in computer science and mathematics through sets, logic, Boolean algebra, proof, counting, recursion, number theory, probability, trees, graphs, and finite-state models.

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Table of Contents

Chapter 1: Discrete Worlds and Sets

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1.1 What Discrete Math Studies
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1.2 Sets, Subsets, and Power Sets
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1.3 Set Operations and Cartesian Products

Chapter 2: Logic and Boolean Algebra

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2.1 Propositions and Logical Connectives
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2.2 Truth Tables, Implication, and Equivalence
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2.3 Predicates and Quantifiers
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2.4 Boolean Algebra, Gates, and Circuits

Chapter 3: Proof

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3.1 Definitions, Conjectures, and Counterexamples
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3.2 Direct Proof, Cases, and Contraposition
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3.3 Contradiction, Existence, and Uniqueness
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3.4 Mathematical and Strong Induction

Chapter 4: Functions, Relations, and Orders

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4.1 Functions, Composition, and Inverses
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4.2 Binary Relations and Their Properties
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4.3 Equivalence Relations and Partitions
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4.4 Partial Orders and Hasse Diagrams

Chapter 5: Algorithms and Growth

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5.1 Describing Algorithms with Pseudocode
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5.2 Correctness and Invariants
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5.3 Counting Steps and Asymptotic Growth

Chapter 6: Counting

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6.1 Sum, Product, and Bijection Principles
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6.2 Permutations and Combinations
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6.3 The Pigeonhole Principle
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6.4 The Binomial Theorem and Inclusion–Exclusion

Chapter 7: Recursion and Recurrences

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7.1 Recursive Definitions and Structural Induction
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7.2 Linear Recurrence Relations
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7.3 Divide-and-Conquer Recurrences and Recursion Trees

Chapter 8: Number Theory and Cryptography

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8.1 Divisibility, Primes, and Factorization
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8.2 GCD, Euclid’s Algorithm, and Bézout’s Identity
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8.3 Modular Arithmetic, Inverses, and the Chinese Remainder Theorem
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8.4 Building a Small RSA System

Chapter 9: Discrete Probability

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9.1 Sample Spaces and Counting Probability
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9.2 Conditional Probability and Independence
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9.3 Bayes’ Theorem
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9.4 Discrete Random Variables and Expected Value

Chapter 10: Trees

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10.1 Tree Properties, Rooted Trees, and Binary Trees
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10.2 Tree Traversals and Expression Trees
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10.3 Prefix Codes, Huffman Trees, and Decision Trees

Chapter 11: Graphs

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11.1 Graph Models, Representations, and Degrees
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11.2 Walks, Paths, Cycles, and Connectivity
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11.3 BFS, DFS, and Shortest Paths
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11.4 Spanning Trees and Minimum Spanning Trees
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11.5 Euler and Hamilton Problems
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11.6 Graph Coloring and Planarity

Chapter 12: Finite-State Models and Capstone

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12.1 Finite-State Machines
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12.2 Capstone: Model and Optimize a Smart-City Network

SUITABLE AUDIENCE

For beginners in computer science, software engineering, mathematics, data science, and related majors, as well as independent learners strengthening their foundations in logic, proof, and discrete structures. Basic algebra is sufficient; calculus and programming are not required.

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