## Introductory Concepts for Abstract MathematicsBeyond calculus, the world of mathematics grows increasingly abstract and places new and challenging demands on those venturing into that realm. As the focus of calculus instruction has become increasingly computational, it leaves many students ill prepared for more advanced work that requires the ability to understand and construct proofs. Introductory Concepts for Abstract Mathematics helps readers bridge that gap. It teaches them to work with abstract ideas and develop a facility with definitions, theorems, and proofs. They learn logical principles, and to justify arguments not by what seems right, but by strict adherence to principles of logic and proven mathematical assertions - and they learn to write clearly in the language of mathematics The author achieves these goals through a methodical treatment of set theory, relations and functions, and number systems, from the natural to the real. He introduces topics not usually addressed at this level, including the remarkable concepts of infinite sets and transfinite cardinal numbers Introductory Concepts for Abstract Mathematics takes readers into the world beyond calculus and ensures their voyage to that world is successful. It imparts a feeling for the beauty of mathematics and its internal harmony, and inspires an eagerness and increased enthusiasm for moving forward in the study of mathematics. |

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### Contents

Logic and Propositional Calculus | 3 |

Tautologies and Validity | 15 |

Quantifiers and Predicates | 29 |

Techniques of Derivation and Rules | 41 |

Techniques | 53 |

Mathematical Induction | 79 |

Sets and Set Operations | 93 |

Set Union Intersection and Complement | 107 |

Composition of Functions | 173 |

Image and Preimage Functions | 179 |

ALGEBRAIC AND ORDER PROPERTIES | 187 |

Finite and Infinite Sets | 237 |

Denumerable and Countable Sets | 243 |

Uncountable Sets | 253 |

Transfinite Cardinal Numbers | 259 |

Partially Ordered Sets | 275 |

Generalized Union and Intersection | 123 |

Cartesian Products | 135 |

Relations | 141 |

Partitions | 151 |

Functions | 161 |

Least Upper Bound and Greatest Lower | 285 |

Well Ordered Sets | 299 |

305 | |

327 | |

### Common terms and phrases

A C B A U B addition and multiplication algebraic argument assertion axiom binary operation cardinally equivalent chain Chapter commutative compound statement conjecture Consequently considered contrapositive corollary countable defined definition denoted derivation method direct proof disjoint equation equivalence relation established example Exercise F F F F T F false finite sets function given Hence implies infinite sets integers intersection inverse least upper bound lemma lower bound mathematical induction maximal element natural numbers negation nonempty set order isomorphism ordered pairs P A Q P V Q partial order relation partially ordered set partition poset predicate preimage premises properties Prove Theorem quantifiers real numbers reflexive respect result Rule E.S. Rule U.G. says set builder notation set theory step student Suppose symbol symmetric tautology transitive trichotomy law true truth table truth value variable Verify Vx[x words