## 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. |

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### Contents

The language of mathematics | 3 |

Implications | 10 |

Proofs | 21 |

Proof by contradiction | 30 |

The induction principle | 39 |

Mathematical statements and proofs | 53 |

Part II | 59 |

The language of set theory | 61 |

The division theorem | 191 |

The Euclidean algorithm | 199 |

Consequences of the Euclidean algorithm | 207 |

Linear diophantine equations | 216 |

Problems IV | 225 |

Part V | 229 |

Congruence of integers | 231 |

Linear congruences | 240 |

Quantifiers | 74 |

Functions | 89 |

Injections surjections and bijections | 101 |

Sets and functions | 115 |

Part III | 121 |

Counting | 123 |

Properties of finite sets | 133 |

Counting functions and subsets | 144 |

Number systems | 157 |

Counting infinite sets | 170 |

Numbers and counting | 182 |

Part IV | 189 |

Congruence classes and the arithmetic of remainders | 250 |

Partitions and equivalence relations | 262 |

Problems V | 271 |

Part VI | 275 |

The sequence of prime numbers | 277 |

Congruence modulo a prime | 289 |

Problems VI | 295 |

Solutions to exercises | 299 |

345 | |

List of symbols | 346 |

347 | |

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### Common terms and phrases

algebraic arithmetic axioms bijection binomial calculation cardinality chapter codomain congruence classes modulo consider Constructing a proof contrapositive coprime counting deduce definition denote diophantine equation disjoint divides division theorem domain equivalence relation Euclidean algorithm example Exercise exist F F F F T F false Fermat's finite sets formal proof formula free variable gcd(a Given Goal greatest common divisor Hence idea implication inclusion-exclusion principle inductive hypothesis inductive step infinite decimal infinite sets injection integer q inverse linear congruence mathematician mathematics maximum element means modular arithmetic modulo multiplication non-empty non-negative integers non-zero notation Notice partition pigeonhole principle positive integers pre-image precisely predicate prime numbers Problems proof by contradiction proof of Proposition properties Prove by induction rational number reader real numbers remainder sequence set of integers set of real simply subset surjection symbol truth table unique well-defined write