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(b style='font-size:24px')(barry31416 style='color:#009900')I. (i)Notions from Set Theory(/i) 1(/barry31416)(/b) ita - chapter 1

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Note: The number in parenthesis at the beginning of each entry indicates the part for which the item was required - and therefore included.

I. Notions from Set Theory 1 ( ita - chapter 1 )

sets, elements, and subsets 2
(1) ____ : Introduction
(1) ____ : Intuitive Description of Sets
(1) ____ : Union of Sets
(1) ____ : Intersection of Sets
(1) ____ : The Null Set
(1) ____ : Material Implication - if A then B
(1) ____ : Subsets

operations on sets 4
(1) ____ : The Compliment of a Set
(1) X 12 : The Compliment of a Union
(1) E 06 : The Compliment of an Intersection
(1) ____ : The Difference of two Sets
(1) ____ : Ordered Pairs
(1) ____ : The Cartesian Product of Two Sets

functions 8
(1) ____ : Function from X to Y
(1) ____ : The General Definition of Function
(1) ____ : The Restriction of a Function
(2) ____ : Composition of Functions
(1) ____ : Inverse Functions
(1) ____ : Image of a Set Under a Function
(1) ____ : Inverse Image of a Set Under a Function

finite and infinite sets 10
(1) ____ : The Natural Numbers
(1) ____ : Finite and Infinite Sets
(1) X 12 : The Image of a Union is the Union of Images
(1) X 12 : The Inverse Image of a Union
(1) X 12 : The Inverse Image of an Image
(1) X 12 : The Image of an Inverse Image


II. The Real Number System 15 ( ita - chapter 2 )

(1) ____ : Introduction
(1) ____ : The Real Number System
(1) ____ : The Field Properties
(1) ____ : The Order Property
(1) ____ : The Least Upper Bound Property
(1) ____ : Main Consequences of the Field Properties
(1) ____ : A Field of Two Elements
(1) ____ : Main Consequences of the Order Property
(1) ____ : The Field Becomes Infinite
(1) ____ : Integral Exponents
(1) ____ : Absolute Values
(1) ____ : Main Consequences of the Least Upper Bound Property
(1) ____ : Summary Page for Upper and Lower Bounds
(1) ____ : Existence of Square Roots
(1) ____ : Bigger Numbers have Bigger Square Roots


III. Metric Spaces 33 ( ita - chapter 3 )

definition of a metric space 34
(1) D 34 : Definition of a Metric Space
(1) E 34 : The Real Numbers as a Metric Space
(1) E 34 : Euclidean n-Space
(1) D 36 : Subspaces in a Metric Space
(1) P 37 : The Difference of Two Sides of a Triangle is Less than the Third Side

open and closed sets 37
(1) D 37 : Open and Closed Balls in a Metric Space
(1) D 38 : Intervals of Real Numbers
(1) D 39 : Open Sets in a Metric Space
(1) P 40 : An Open Ball is an Open Set
(1) D 40 : Closed Sets in a Metric Space
(1) P 41 : A Closed Ball is a Closed Set
(1) ____ : Examples with Real Numbers
(1) P 39 : Combinations of Open Sets
(1) P 41 : Combinations of Closed Sets
(1) E 42 : Half Spaces in Euclidean n-Space
(1) E 43 : Intervals in Euclidean n-Space
(1) D 43 : Bounded Sets
(1) P 44 : Extrema for Nonempty Bounded Closed Sets of Real Numbers
(1) X 61 : Bounded Open Sets of Real Numbers

convergent sequences 44
(1) D 45 : Convergence of a Sequence
(1) E 46 : The Constant Sequence
(1) P 46 : The Limit of a Sequence is Unique
(1) D 46 : Subsequences
(1) P 46 : Convergence of Subsequences
(1) D 47 : Bounded Sequences
(1) E 47 : Convergent Sequences are Bounded
(1) T 47 : Convergent Sequences and Closed Sets
(1) P 48 : Limit of a Product Sequence
(1) P 48 : Limits of Sum, Difference, and Quotient Sequences
(1) D 50 : Monotonic Sequences
(1) P 50 : Bounded Monotonic Sequences are Convergent

completeness 51
(1) D 51 : Cauchy Sequences
(1) P 51 : Convergent Sequences are Cauchy
(1) P 52 : Cauchy Sequences are Bounded
(1) P 52 : Convergence of a Cauchy Sequence with a Convergent Subsequence
(1) D 52 : Completeness
(1) P 52 : The Completeness of a Closed Subset of a Complete Metric Space
(1) T 53 : As a Metric Space, the Real Numbers are Complete

compactness 54
(1) D 54 : Compactness
(1) P 54 : A Compact Subset of a Metric Space is Bounded
(1) D 55 : Cluster Points
(1) ____ : Cluster Points in Open Sets of Real Numbers
(1) T 56 : Existence of a Cluster Point in Infinite Subsets of a Compact Metric Space
(1) C 56 : Convergence of a Subsequence in a Compact Metric Space
(1) C 56 : A Compact Metric Space is Complete
(1) C 56 : A Compact Subset of a Metric Space is Closed
(1) X 64 : [a,b] is Compact
(1) X 65 : More on Compactness

connectedness 59
(1) D 59 : Connectedness
(1) P 60 : A Criterion for a Set of Real Numbers to be Not Connected
(1) T 60 : Intervals of Real Numbers are Connected


IV. Continuous Functions 67 ( ita - chapter 4 )

definition of continuity 68
(1) D 68 : Continuity at a Point
(1) D 68 : Continuity on a Set
(1) ____ : Example: f(x,y) = y
(1) E 69 : A Constant Function
(2) P 70 : The Identity Function is Continuous
(1) E 70 : The Restriction of a Continuous Function is Continuous
(1) P 70 : Inverse Images of Open Sets Under Continuous Functions are Open
(2) P 71 : A Continuous Function of a Continuous Function is Continuous

continuity and limits 72
(1) E & D 72 : Limit of a Function at a Point
(1) E 74 : The Connection Between Continuity and Limits
(1) P 74 : Continuous Functions and Convergent Sequences

continuity of rational operations 75
(1) P 75 : Continuity of Rational Operations
(1) >C 76 : Limits in Rational Operations

continuous functions on a compact metric space 78
(1) T 78 : The Continuous Image of a Compact Metric Space is Compact
(1) C 78 : Max & Min - Continuous Real-valued Functions on Compact Metric Spaces
(1) D 80 : Uniform Continuity
(1) E 80 : Uniform Continuity Implies Continuity
(1) T 80 : Continuous Functions on Compact Metric Spaces are Uniformly Continuous

continuous functions on a connected metric space 82
(1) T 82 : The Continuous Image of a Connected Set is Connected
(1) C 82 : The Intermediate Value Theorem

sequences of functions 83
(1) D 83 : Convervent Sequences of Functions
(1) D 85 : Uniformly Convervent Sequences of Functions
(1) P 86 : Cauchy Criterion for Sequences of Functions
(1) T 87 : Continuity of the Limit of a Uniformly Convergent Sequence of Continuous Functions
(1) L 87 : Continuity of f,g: E to E' Implies Continuity of d'( f(p), g(p) )
(1) T 90 : A Complete Metric Space of Continuous Functions


V. Differentiation 97 ( ita - chapter 5 )

definition of derivative 98
(1) D 098 : Differentiability at a Point
(1) P 099 : Differentiability Implies Continuity
(1) D 100 : Differentiability on a Set
(1) ____: The Restriction of a Differentiable Function is Differentiable

rules of differentiation 100
(1) E 100 : The Derivative of a Constant Function is Zero
(1) E 100 : The Derivative of the Identity Function is One
(1) P 101 : Rules of Differentiation
(1) ____: The Derivative of a Product of n Functions
(2) P 103 : The Chain Rule

the mean value theorem 103
(1) P 103 : The Derivative at Extrema is Zero
(1) L 104 : Rolle's Theorem
(1) T 105 : The Mean Value Theorem
(2) D 105 : Increasing and Decreasing Functions
(2) C 105 : Increasing and Decreasing Functions - The Derivative's Implication


VI. Riemann Integration 111 ( ita - chapter 6 )

definitions and examples 112
(1) D 112 : The Riemann Sum
(1) D 112 : The Riemann Integral
(1) E 114 : Integral of a Constant Function
(1) E 114 : Integral of a Function with Only One Non-zero Point
(1) E 114 : Integral of a Unit Step

linearity and order properties of the integral 116
(1) P 116 : Integrals of Sums and Constant Multiples of Functions
(1) P 117 : The Integral of a Non-negative Function is Non-negative
(1) C 117 : Integration of an Inequality
(1) C 118 : Bounds for an Integral

existence of the integral 118
(1) L 118 : Integrability
(1) D 119 : Step Functions
(1) L 119 : Step Functions are Integrable
(1) P 120 : General Condition for Riemann Integrability
(1) T 123 : Continuous Functions are Riemann Integrable

the fundamental theorem of calculus 123
(1) P 123 : Subintervals in Riemann Integration
(1) D 125 : Negation of a Riemann Integral
(1) C 125 : Integration from a to b to c - Regardless of Order
(1) T 126 : The Fundamental Theorem of Calculus
(1) C 127 : The Integral in Terms of its Antiderivative

the logarithmic and exponential functions 128
(2) D 128 : Definition of the Logarithm
(2) P 128 : Properties of the Logarithm Function
(2) P 129 : Definition of the Exponential Function


VII. Interchange of Limit Operations 137 ( ita - chapter 7 )

differentiation under the integral sign 159
(1) D 159 : The Partial Derivative


VIII. The Method of Successive Approximations 169 ( ita - chapter 8 )

the fixed point theorem 170
(1) T 170 : The Fixed Point Theorem

uniqueness of the real number system 29
(1) ____: b-mal Expansions of Real Numbers
(1) ____: Every Infinite b-mal Represents a Real Number
(1) ____: Periodic Infinite b-mals
(1) Discussion 29 : Uniqueness of the Real Number System

the simplest case of the implicit function theorem 173
(1) T 174 : The Simplest Case of The Implicit Function Theorem
(1) C 176 : The Inverse Function Theorem

existence and uniqueness theorems for ordinary differential equations 177
(1) T 178 : dy/dx = f(x,y)


IX. Partial Differentiation 193 ( ita - chapter 9 )


X. Multiple Integrals 215 ( ita - chapter 10 )