Discrete Mathematics and Its Applications

Citation

Rosen, K. H. (2025). Discrete Mathematics and Its Applications (2025 release). McGraw Hill.

Chapter Summary

McGraw Hill lists thirteen chapters followed by appendices for the 2025 release; the publisher's contents for the seventh edition list the same thirteen chapter topics. The public contents give chapter titles only, so details below that come from the publisher's change list for the eighth edition (2019) or from the preface of the seventh edition (2012) are attributed to those documents.

Chapters 1–2: Foundations: Logic and Proofs; Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Chapter 1 starts the book with logic and then proof methods. The publisher's eighth-edition change list mentions logical puzzles and an example that models the n-queens problem as a satisfiability problem. Chapter 2 gathers the basic structures used throughout: sets, functions, sequences, sums and matrices; the same list records that multisets moved from the exercises into the text.

Chapters 3–4: Algorithms; Number Theory and Cryptography

Chapter 3 treats algorithms; according to the eighth-edition change list it includes the string matching problem and its brute-force algorithm. Chapter 4 covers number theory and cryptography; that list mentions the extended Euclidean algorithm, recent discoveries about primes and open conjectures, and homomorphic encryption.

Chapters 5–6: Induction and Recursion; Counting

Chapter 5 is devoted to induction and recursion; the eighth-edition changes include an expanded template for proofs by mathematical induction, placed before the worked examples. Chapter 6 introduces counting, and the same list records expanded coverage of the division rule.

Chapters 7–8: Discrete Probability; Advanced Counting Techniques

Chapter 7 covers discrete probability and Chapter 8 counting techniques that go beyond the basics. Neither is described in more detail in the public contents.

Chapters 9–11: Relations; Graphs; Trees

Chapter 9 treats relations; the eighth-edition change list notes that association rules from data mining are discussed in the section on n-ary relations. Chapter 10 on graphs includes, according to the same list, a new example that models semantic networks from artificial intelligence as graphs. Chapter 11 turns to trees.

Chapters 12–13: Boolean Algebra; Modeling Computation

The last two chapters cover Boolean algebra and the modelling of computation. The seventh-edition preface marks both as optional for computer science emphasis rather than core reading.

Appendices

The seventh-edition preface describes three appendixes: axioms for the real numbers and positive integers, exponential and logarithmic functions, and the pseudocode used to express algorithms. The eighth-edition change list adds that a chapter on abstract algebra (groups, rings and fields) is available as a separate online resource that can be added to custom versions.

Key Concepts

Chapters 1–6: Logic, structures, algorithms, number theory, counting

• Logic and proofs: the foundation named in the Chapter 1 title, which the preface calls the base for the later discussion of proof methods.
• Sets, functions, sequences, sums and matrices: the basic structures of Chapter 2.
• Algorithms: the subject of Chapter 3; the preface says algorithms are written in words and in pseudocode, and that their computational complexity is analysed at an elementary level.
• Number theory and cryptography: the subject of Chapter 4.
• Induction and recursion: Chapter 5, with mathematical induction stressed in the preface.

Chapters 7–13: Probability, relations, graphs, trees and computation

• Discrete probability and advanced counting: Chapters 7–8.
• Relations: Chapter 9.
• Graphs and trees: Chapters 10–11.
• Boolean algebra and modeling computation: Chapters 12–13.

Critical Analysis

Strengths

• Breadth in one volume. The contents range from logic and proofs to graphs, Boolean algebra and models of computation, and the publisher states that the book was written for mathematics, computer science and engineering majors, so one text can serve several kinds of course.
• Stated pedagogical themes. The seventh-edition preface names five threads that run through the book: mathematical reasoning, combinatorial analysis, discrete structures, algorithmic thinking, and applications and modelling. It reports over 800 worked examples, over 4,000 exercises graded by difficulty with answers or outlines to odd-numbered ones, key-terms lists and review questions after each chapter, and optional computer projects and writing projects (figures for that edition).
• Designed for flexible courses. The same preface says the dependence of chapters on each other is minimised and provides a table of core and optional sections, so a one-term sophomore course, a two-term course and a computer science-oriented course can each select different parts.
• Explicit prerequisites. The preface gives college algebra as the only explicit prerequisite and says the few places that refer to calculus are marked, which makes the book usable without a calculus course.

Limitations

• Little detail in the public contents. The publisher shows only chapter titles for the 2025 release and does not itemise what changed from the eighth edition; anyone choosing between editions should compare the actual contents.
• Proofs are a means rather than the central subject. The preface presents mathematical reasoning as one of five themes among others. Students who want a more sustained treatment of how to write proofs may prefer a text whose organising principle is proof-writing, as the Epp book in this course list is.
• Mixed audience. Because the book serves mathematics, computer science and engineering readers, some chapters (Algorithms, Boolean Algebra, Modeling Computation) are marked optional for a mathematics-oriented course in the seventh edition's table, while Chapters 5, 6, 9 and 10 hold sections that the table treats as core for every course.

Where It Fits in the Course

Rosen is a broad, reference-style text, covering proof methods, algorithms, number theory, counting, probability, graphs and models of computation in a single volume. Epp's Discrete Mathematics with Applications covers similar ground but is organised around learning to reason and write proofs, so it suits students who need more help with logic and proof writing. Oscar Levin's Discrete Mathematics: An Open Introduction is a free, inquiry-based first course built around counting, sequences, graph theory and logic, and László Lovász, József Pelikán and Katalin Vesztergombi's Discrete Mathematics: Elementary and Beyond is a short, problem-driven introduction. Graham, Knuth and Patashnik's Concrete Mathematics and van Lint and Wilson's A Course in Combinatorics come later: the first for sums, recurrences, binomial coefficients and generating functions in the analysis of algorithms, the second as a graduate-level combinatorics text.

How to Use This Book

Rosen suits first- or second-year students in computer science, mathematics and engineering taking a one- or two-term discrete mathematics course; the preface requires only college algebra. It works best as a course text with a reference role afterwards.

A practical plan:

• Core: work through Chapter 1 and the sections on proof methods first, then the basic structures of Chapter 2, induction and recursion (Chapter 5), counting (Chapter 6) and graphs (Chapter 10). The seventh-edition table marks most of Chapters 1, 2, 4, 5, 6, 9 and 10 as core.
• Algorithms and number theory (Chapters 3–4): the seventh-edition table treats Chapter 3 as optional computer science material and only the opening sections of Chapter 4 as core; a computer science course will want both chapters in full.
• Probability, advanced counting and relations (Chapters 7–9): the seventh-edition table treats only one to three sections of each as core, so a short course can take those and a two-term course the rest.
• Optional (Chapters 11–13): the seventh-edition table treats only the opening section on trees as core and Chapters 12–13 (Boolean algebra, modeling computation) as optional computer science material, so they suit a course with that emphasis.

Use the exercise sets steadily: the preface says answers or solutions are given for odd-numbered exercises, and the end-of-chapter review questions are meant to be answered in full sentences rather than with a quick calculation.