Transcription of An Introduction to Higher Mathematics
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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.
Just as in algebra, where multiplication takes precedence over addition, we can eliminate some parentheses by agreeing on a particular order in which logical ... Observe how the inclusion of intermediate steps makes the table easier to calculate and read. A tautology is a logical expression that always evaluates to T, that is, the last column ...
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