An Introduction to Mathematical Reasoning: Numbers, Sets and FunctionsThis book eases students into the rigors of university mathematics. The emphasis is on understanding and constructing proofs and writing clear mathematics. The author achieves this by exploring set theory, combinatorics, and number theory, topics that include many fundamental ideas and may not be a part of a young mathematician's toolkit. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the all-time-great classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. The over 250 problems include questions to interest and challenge the most able student but also plenty of routine exercises to help familiarize the reader with the basic ideas. |
Contents
Sets and functions | 59 |
Numbers and counting | 121 |
Arithmetic | 189 |
Modular arithmetic | 229 |
Prime numbers | 275 |
Solutions to exercises | 299 |
Bibliography | 345 |
| 346 | |
| 347 | |
Common terms and phrases
algebraic arithmetic axioms bijection binomial calculation cardinality Cartesian product chapter codomain congruence classes modulo consider Constructing a proof contrapositive coprime counting deduce definition denote diophantine equation disjoint divides division theorem equivalence relation Euclidean algorithm example Exercise exist false Fermat's finite sets formal proof formula free variable function f gcd(a,b Given Goal gives greatest common divisor Hence idea implication inductive hypothesis inductive step infinite decimal injection integer q inverse linear congruence mathematician mathematics means modular arithmetic modulo multiplication non-empty non-negative integers non-zero notation Notice number theory partition pigeonhole principle positive integers pre-image predicate prime numbers Problems proof by contradiction proof of Proposition properties rational number reader real numbers remainder sequence set of integers simply subset surjection symbol true truth table unique well-defined write Zºt