m3111 - Math 311–03 Advanced Calculus Wolmer V...

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Unformatted text preview: Math 311–03: Advanced Calculus Wolmer V. Vasconcelos Set 1 Spring 2008 Wolmer Vasconcelos (Set 1) Advanced Calculus Spring 2008 1 / 77 Outline 1 General Orientation 2 Rational Numbers 3 Basic Set Theory 4 R : Completeness 5 Cardinality and Countability 6 Last Time & Today 7 Workshop #1 8 Cardinality of R 9 Cantor’s Universe 10 Workshop #2 Wolmer Vasconcelos (Set 1) Advanced Calculus Spring 2008 2 / 77 General Orientation • Pre-requisites: Calc 4, Math 300 • web:www.math.rutgers.edu/(tilde)vasconce • Meetings: MWTh 1:40-3:00 SEC-205 • Office Hours [Hill 228]: MW 11-12, or by arrangement • Textbook: Introduction to Analysis , 5th Ed., by E. D. Gaughan • All this detailed in General Info page: Look over Wolmer Vasconcelos (Set 1) Advanced Calculus Spring 2008 3 / 77 Scoring Info • Quizzes Total: 50 • Workshops Total: 100 • 2 Midterms Total: 2 x 100 = 200 • Final: 200 • Total: 550 pts Wolmer Vasconcelos (Set 1) Advanced Calculus Spring 2008 4 / 77 Some Goals What is R , and what are some of its important properties? Topology of R : continuous functions Really Understand objects such Z b a f ( x ) dx a 1 + a 2 + a 3 + ··· Wolmer Vasconcelos (Set 1) Advanced Calculus Spring 2008 5 / 77 Outline 1 General Orientation 2 Rational Numbers 3 Basic Set Theory 4 R : Completeness 5 Cardinality and Countability 6 Last Time & Today 7 Workshop #1 8 Cardinality of R 9 Cantor’s Universe 10 Workshop #2 Wolmer Vasconcelos (Set 1) Advanced Calculus Spring 2008 6 / 77 Rational Numbers At the outset of our journey are the natural numbers N = { 1 , 2 , 3 , 4 ,... } Its ‘modern’ construction [e.g. Peano’s] is a paradigm of beauty. It is enlarged by the integers N ⊂ Z = { ...,- 4 ,- 3 ,- 2 ,- 1 , , 1 , 2 , 3 , 4 ,... } and the rational numbers N ⊂ Z ⊂ Q = n m n , m , n ∈ Z , n 6 = o These sets exhibit different structures : of a monoid, of a ring and of a field, respectively. Wolmer Vasconcelos (Set 1) Advanced Calculus Spring 2008 7 / 77 Peano The construction by Peano of the set N is grounded on two ingredients: The set N contains a particular element 1. • [Successor Function] There is a function s : N → N that is injective, and for every n ∈ N s ( n ) 6 = 1. • [Induction Axiom] If the subset S ⊂ N has the properties 1 ∈ S & whenever n ∈ S ⇒ s ( n ) ∈ S then S = N Wolmer Vasconcelos (Set 1) Advanced Calculus Spring 2008 8 / 77 Given these definitions, we can define several operations/compositions and structures on N : a + b :=? a + 1 := s ( a ) a + s ( n ) := s ( a + n ) a × b :=? a × 1 := a a × s ( n ) := a × n + a Wolmer Vasconcelos (Set 1) Advanced Calculus Spring 2008 9 / 77 Example Theorem Suppose a ≥ - 1 . Then for all n ∈ N , ( 1 + a ) n ≥ 1 + na....
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m3111 - Math 311–03 Advanced Calculus Wolmer V...

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