## 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 |

List of symbols | |

### Other editions - View all

An Introduction to Mathematical Reasoning: Numbers, Sets and Functions Peter J. Eccles Limited preview - 2013 |

An Introduction to Mathematical Reasoning: Numbers, Sets and Functions Peter J. Eccles Limited preview - 1997 |

An Introduction to Mathematical Reasoning: Numbers, Sets and Functions Peter J. Eccles No preview available - 1997 |

### Common terms and phrases

addition apply argument arithmetic axioms Base bijection calculation called cardinality chapter congruence classes consider Constructing a proof contradiction coprime corresponds counting deduce defined definition denote denumerable described determine divides divisible element equal equivalence Euclidean algorithm example Exercise exist fact factorization false finite sets formal formula function given gives Goal greatest common divisor Hence holds idea illustrates implication indicate inductive hypothesis inductive step injection inverse involving leads mathematics means method modulo multiplication necessary non-negative notation Notice observe obtain positive integers possible precisely predicate prime prime numbers principle Problems proof properties Proposition prove provides Question rational number reader real numbers relation remainder represent result sequence simply solution Solve square statement subset Suppose surjection symbol theorem true unique universal usually write written