Discrete Mathematics for Computer Science, Some Notes

These are notes on discrete mathematics for computer scientists. The presentation is somewhat unconventional. Indeed I begin with a discussion of the basic rules of mathematical reasoning and of the notion of proof formalized in a natural deduction system ``a la Prawitz’’. The rest of the material is more or less traditional but I emphasize partial functions more than usual (after all, programs may not terminate for all input) and I provide a fairly complete account of the basic concepts of graph theory.

💡 Research Summary

The manuscript “Discrete Mathematics for Computer Science, Some Notes” by Jean Gallier is a comprehensive set of lecture notes aimed at undergraduate computer‑science students. Its overarching goal is to present the core topics of discrete mathematics while explicitly linking them to the needs of program correctness, algorithm design, and formal reasoning.

The work opens with a pedagogical justification for teaching discrete mathematics to CS majors, emphasizing that many students lack a clear notion of proof and that rigorous reasoning is essential for software verification. Gallier proposes to start with the formalization of proof itself, using a natural‑deduction system “à la Prawitz”. He introduces two proof systems, N ⇒ m and NG ⇒ m, and systematically presents inference rules for conjunction, disjunction, falsum (⊥), truth (⊤), universal and existential quantifiers, as well as rules for ⊥‑elimination and ⊥‑introduction. The treatment includes a brief discussion of normalization, decision procedures, and the relationship between truth and provability, but deliberately avoids deep semantic issues, focusing instead on the syntactic structure of deductions.

Chapter 2 shifts to relations, functions, and especially partial functions, a theme that recurs throughout the notes. Gallier argues that partial functions model non‑terminating programs and therefore deserve explicit treatment. He defines ordered pairs, Cartesian products, and the various properties of functions (injectivity, surjectivity, bijectivity). The chapter covers composition, recursion on ℕ, inverses, direct and inverse images, the pigeon‑hole principle, the Schröder–Bernstein theorem, and even Hilbert’s space‑filling curve as an “amazing surjection”. Strings, multisets, and indexed families are introduced as data structures familiar to programmers.

Chapter 3 is a compact combinatorics module. It treats counting permutations and functions, binomial coefficients, multinomial coefficients, and the inclusion‑exclusion principle, providing proofs that rely on the induction and recursion machinery developed earlier.

Chapter 4 deals with order‑theoretic concepts. After defining partial orders, Gallier presents lattices, Tarski’s fixed‑point theorem, well‑founded orderings, and complete induction. He uses these tools to prove unique prime factorization in ℤ and to discuss greatest common divisors. The section also covers equivalence relations, partitions, transitive and reflexive closures, distributive lattices, Boolean algebras, and Heyting algebras. These topics are labeled as “advanced” and may be omitted in a core curriculum.

Chapter 5 is the most extensive part of the notes and constitutes a “fairly complete” treatment of graph theory. It begins with motivations for studying graphs, then proceeds through directed graphs, paths, strongly connected components, undirected graphs, chains, cycles, connectivity, trees, spanning trees (including minimum‑weight versions), Γ‑cycles, cocycles, cotrees, flows and tensions, incidence and adjacency matrices, Eulerian and Hamiltonian cycles, network‑flow problems, the Max‑Flow Min‑Cut theorem (presented via a proof due to M. Sakharovitch), matchings, coverings, bipartite graphs, and planar graphs. The exposition balances formal definitions with illustrative examples, and the Max‑Flow Min‑Cut proof showcases the author’s preference for constructive, algorithm‑oriented arguments.

Throughout the manuscript Gallier interleaves historical remarks, references to classic texts (van Dalen, Huth & Ryan, Troelstra & Schwichtenberg), and practical advice for instructors (e.g., which chapters can be skipped, how to adapt the material for different student backgrounds). He acknowledges that the logic‑first approach may be challenging for some learners but reports that a majority of his own students found it engaging because of its similarity to programming.

In summary, the notes provide a logically rigorous yet computer‑science‑oriented pathway through discrete mathematics. By grounding proof theory in natural deduction, emphasizing partial functions, and delivering a thorough graph‑theoretic toolkit, the work bridges the gap between abstract mathematics and concrete algorithmic practice. Its strengths lie in the unified proof‑centric perspective and the explicit connection to program semantics; its weaknesses are the occasional depth that may overwhelm beginners and the limited number of worked programming‑style examples. Nonetheless, it constitutes a valuable resource for instructors seeking a cohesive, proof‑focused syllabus for a modern CS discrete‑math course.