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An Introduction to Higher Mathematics

An Introduction to Higher Mathematics Patrick Keef David Guichard with modifications by Russ Gordon Whitman College c 2010. Contents 1. Logic 1. Logical Operations .. 1. George Boole .. 6. Quantifiers .. 8. De Morgan's Laws .. 11. Augustus De Morgan .. 13. Mixed Quantifiers .. 15. Logic and Sets .. 17. Ren e Descartes .. 20. Families of Sets .. 22. Equivalence Relations .. 24. 2. Proofs 29. Direct Proofs .. 32. Divisibility .. 35. Existence proofs .. 37. Mathematical Induction .. 40. Two Important Results .. 44. Strong Induction .. 49. Well-Ordering Property .. 53. Indirect Proof .. 58. Euclid of Alexandria .. 60. iii iv Contents 3. Number Theory 63. Congruence .. 63. Carl Friedrich Gauss .. 66. The spaces Zn.

of mathematics; for our purposes, a brief introduction will give us the means to investigate more traditional mathematics with con dence. 1.1 Logical Operations Mathematics typically involves combining true (or hypothetically true) statements in various ways to produce (or prove) new true statements. We begin by clarifying some of these fundamental

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