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

Learn discrete mathematics as a method for reasoning formally Are 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. Read more