## Foundations of higher mathematicsThis text introduces students to basic techniques of writing proofs and acquaints them with some fundamental ideas. The authors assume that students using this text have already taken courses in which they developed the skill of using results and arguments that others have conceived. This text picks up where the others left off -- it develops the students' ability to think mathematically and to distinguish mathematical thinking from wishful thinking. |

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### Contents

THE LOGIC AND LANGUAGE OF PROOFS | 1 |

SETS | 31 |

MATHEMATICAL INDUCTION | 51 |

Copyright | |

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### Common terms and phrases

accumulation point Algebra Axiom of Choice belongs called Cauchy sequence Chapter complex numbers congruence classes consider continuous function contrapositive converges countably infinite Dedekind infinite definition denote directed graph divides divisor element equivalence classes equivalence relation exactly exists false Figure finite set Give an example given greatest lower bound Hint infinite set infinite subset integer integral domain inverse isomorphic least member Lemma Let G Mathematical Induction metric space multiplication natural number nonempty set nonempty subset notation one-to-one function one-to-one function mapping operation ordered integral domain ordered pairs partial order partition permutations positive number positive real numbers Principle of Mathematical PROOF Let proof of Theorem PROOF See Exercise Prove Proposition Prove Theorem Prove your answer rational numbers reader real number real-valued functions set theory smallest member square truth table uniformly continuous Wayne is playing