xxxxx - MATHEMATICS 300 FALL 2006 Introduction to...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATHEMATICS 300 FALL 2006 Introduction to Mathematical Reasoning H. J. Sussmann INSTRUCTORS NOTES Pages 1 to 62 (September 14, 2006) Contents 1 Information on the course 1 1.1 Course schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 About the instructor . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Web page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.4 Office hours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.5 Lectures and exams . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.6 Your final grade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.7 The textbook and the instructors notes . . . . . . . . . . . . . . . 2 1.8 Always bring the book to class! . . . . . . . . . . . . . . . . . . . . 2 1.9 Readings for the first 4 days (September 5, 7, 12 and 14) . . . . . . 3 1.10 Homework assignment no. 1, due on Thursday September 14 . . . 3 1.11 Some remarks about mathematical writing . . . . . . . . . . . . . . 4 1.11.1 Write clearly in complete sentences . . . . . . . . . . . . . . 4 1.11.2 Your written work . . . . . . . . . . . . . . . . . . . . . . . 6 1.12 Answering questions in this course . . . . . . . . . . . . . . . . . . 7 1.13 Some examples of problems with correct answers . . . . . . . . . . 8 2 First remarks on reasoning and proofs 11 2.1 Reasoning and arguments . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1 Some examples of good and bad arguments . . . . . . . . . 15 2.1.2 An argument part of a dialogue between an advocate and a skeptic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.3 Arguments must be sequential . . . . . . . . . . . . . . . . 21 2.1.4 Circular arguments are completely invalid . . . . . . . . . . 23 2.1.5 When is an argument convincing? . . . . . . . . . . . . . . 25 2.1.6 The Same Kind of Argument principle. . . . . . . . . . . 31 3 A first look at the mathematical zoo: number systems 38 ii Sussmann Math 300 Fall 2006 3.1 Numbers: IN, Z , Z + , Q and IR. . . . . . . . . . . . . . . . . . . . . 39 3.2 Reading formulas with and . . . . . . . . . . . . . . . . . . . . 42 3.3 The complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.4 The integers modulo n . . . . . . . . . . . . . . . . . . . . . . . . . 44 4 Mathematical proofs 53 4.1 Some examples of mathematical proofs . . . . . . . . . . . . . . . . 54 4.1.1 A first list of inference rules . . . . . . . . . . . . . . . . . . 54 4.1.2 A first list of known facts from arithmetic . . . . . . . . . . 56 4.1.3 Proof that the equation x 2 +1 = 0 does not have a real solution 56 4.1.4 A second list of inference rules . . . . . . . . . . . . . . . . 57 4.1.5 Some definitions . . . . . . . . . . . . . . . . . . . . . . . . 58 4.1.6 A second list of known facts . . . . . . . . . . . . . . . . . . 58 4.1.7 Proof that 2 is irrational . . . . . . . . . . . . . . . . . . . 59 4.1.8 Proof that the sum of two even integers is even . . . . . . .Proof that the sum of two even integers is even ....
View Full Document

This note was uploaded on 01/31/2011 for the course MATH 300 taught by Professor Ctw during the Spring '08 term at Rutgers.

Page1 / 64

xxxxx - MATHEMATICS 300 FALL 2006 Introduction to...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online