Transcription of An Introduction to Higher Mathematics - Whitman People
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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 .. 68. The Euclidean Algorithm .. 71. The spaces Un .. 75. The GCD and the LCM.
2 Chapter 1 Logic since the rst is a matter of opinion and the second leads to a logical dilemma. More generally, by a formula we mean a statement, possibly involving some variables, which is …
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