The Discrete Mathematics Method: Concepts, Worked Examples, and Exercises to Master Logic, Proofs, Graphs, and Algorithms

Learn discrete mathematics as a method for reasoning formallyAre there more real numbers than integers, even though both sets are infinite? Cantor's diagonal argument answers with striking elegance: suppose you could list every real number between 0 and 1, then construct a new real that differs from the first in its first digit, from the second in its second digit, and so on. This new real appears in no row — your list must have been incomplete. Not all infinities are equal. This result, at the heart of Chapter 4, transforms the way you think about the infinite. This book takes you from propositional logic all the way to complexity analysis — by hand, without software, through rigorous reasoning alone.Across 8 progressive chapters, you discover propositional logic and predicate calculus with truth tables and quantifiers, three proof techniques (direct proof, contrapositive, induction) with a 4-step induction recipe, sets and functions with injectivity, surjectivity, bijectivity, and Cantor's diagonal argument (the WOW chapter), modular arithmetic and RSA encryption rebuilt by hand from first principles to decryption, combinatorics with binomial coefficients, the pigeonhole principle, and inclusion-exclusion, graph theory with breadth-first search, Euler's circuit theorem, and minimum spanning trees, and algorithmic analysis with big-O notation, recurrences, and the master theorem. Three recipes in 4 or 5 steps structure the method: the induction recipe, the breadth-first-search recipe, and the complexity-analysis recipe.What's inside8 progressive chapters, from propositional logic to algorithmic analysis25 worked examples in full detail to anchor every technique72 exercises with complete, step-by-step solutions3 structuring recipes: induction, breadth-first search, complexity analysisCantor's diagonal argument: the proof that reveals not all infinities are equalRSA encryption rebuilt by hand, from choosing primes to full decryptionEuler's circuit theorem, from the Königsberg bridges to the parity-of-degrees conditionWhy this bookA method, not a treatise: you learn to reason, not to memorize definitionsThe three structuring recipes: the tools every computer scientist and mathematician actually usesFour-step pedagogy: concept, example, exercise, solutionHand computation exclusively: no software required, understanding guaranteedSuited for self-study, undergraduate and graduate programs in CS, math, or engineeringWhether you are pursuing a degree in computer science, mathematics, or engineering, a professional refreshing your formal-reasoning foundations, or a self-learner who wants to understand the foundations of algorithms beyond the code — this book is your method. A few chapters are enough to transform the way you think about discrete structures and algorithmics. Read more