Discrete Mathematics (Google eBook)
This books gives an introduction to discrete mathematics for beginning undergraduates. One of original features of this book is that it begins with a presentation of the rules of logic as used in mathematics. Many examples of formal and informal proofs are given. With this logical framework firmly in place, the book describes the major axioms of set theory and introduces the natural numbers. The rest of the book is more standard. It deals with functions and relations, directed and undirected graphs, and an introduction to combinatorics. There is a section on public key cryptography and RSA, with complete proofs of Fermat's little theorem and the correctness of the RSA scheme, as well as explicit algorithms to perform modular arithmetic. The last chapter provides more graph theory. Eulerian and Hamiltonian cycles are discussed. Then, we study flows and tensions and state and prove the max flow min-cut theorem. We also discuss matchings, covering, bipartite graphs.
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algorithm assume axiom of choice axioms bijection binomial bipartite graph called chain classical logic cocycle complete induction connected components contradiction deduction tree define Definition denoted directed graph elimination rule endpoints equivalence relation Euler example fact Fibonacci finite set first-order flow f formula function f given graph G Heyting algebra holds incidence matrix induction hypothesis inference rules infinite injective intuitionistic logic inverse lattice Let G maximum matching minimal modulo multiset natural numbers nodes nonempty notation notion partial order path pigeonhole principle planar plane graph poset prime numbers problem proof system proof tree proof-by-contradiction rule properties Proposition prove PV Q recursively relatively prime Remark Section sequence set of edges set of premises shown in Figure simple cycle spanning tree strongly connected strongly connected components subgraph subset surjective Theorem truth assignment undirected unique valid variable vector vertex vertices