## 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

SETS | 37 |

MATHEMATICAL INDUCTION | 59 |

COMBINATORIAL PROOFS | 95 |

Copyright | |

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### Common terms and phrases

a e G Algebra assume Axiom of Choice belongs Binomial called cardinal number chapter complex number congruence classes consider contrapositive countable set countably infinite counterexample defined denote directed graph disjoint divides domain equation equivalence classes equivalence relation exactly exists false Figure Find finite set formula given greatest lower bound Hint homomorphism infinite set integer isomorphic least member Let G let h Let x e Mathematical Induction multiplication natural number nonempty set notation number greater number n odd integer one-to-one function one-to-one function mapping operation ordered pairs partial order partition permutations Pigeonhole Principle positive integer positive real numbers prime number Principle of Mathematical problem PROOF Let PROOF See Exercise propositional function Prove by induction Prove Proposition Prove Theorem Q is true rational numbers real number reflexive Schroeder-Bernstein Theorem sequence set theory square subgroup of G symmetric term vertices Zermelo-Fraenkel Set Theory