Difference between revisions of "MAT3213"

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(Topic list Weeks 1&2.)
(Topics until Week 4.)
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|| <!--Topics-->   
 
|| <!--Topics-->   
  
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[[NaturalNumbers:Well-Ordering]]
+
[[NaturalNumbers:Well Ordering]]
 
[[Proofs:Induction]]
 
[[Proofs:Induction]]
 
[[Induction:Variants]]
 
[[Induction:Variants]]
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|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
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* [[Set Operations]] <!--- 3213-1.1 --->
+
[[Sets:Operations]]
−
* [[Injective and Surjective Functions]] <!--- 3213-1.1 --->
+
[[NaturalNumbers:Well Ordering]]
 +
[[Proofs:Induction]]
 +
[[Functions:Injective]]
 +
[[Functions:Bijective]]
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
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|| <!--Sections-->  
 
|| <!--Sections-->  
  
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2.1
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2.1-2.2
  
 
|| <!--Topics-->   
 
|| <!--Topics-->   
  
−
[[Algebraic Properties of the Real Numbers]]
+
[[Real Numbers]]
 +
[[RealNumbers:Algebraic Properties]]
 +
[[RealNumbers:Order]]
 +
[[RealNumbers:Inequalities|Inequalities]]
 +
[[RealNumbers:Absolute Value|Absolute Value]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* '''[[Field Properties]]''' <!--- DNE (recommend Modern Algebra) --->
+
* [[Functions:Operations]] <!--- DNE (recommend Modern Algebra) --->
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
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|| <!--Sections-->  
 
|| <!--Sections-->  
  
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2.1
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2.3-2.4
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|| <!--Topics-->
 
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[[Rational and Irrational Numbers]]
 
−
 
 
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|| <!--Prerequisite Skills-->
 
−
 
 
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* [[Restrictions on Functions| The square root function]] <!--- 3213-1.1 --->
 
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* [[Functions(The Cartesian Product Definition)]] <!--- 3213-1.1 --->
 
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* '''[[Definition of Even and Odd Numbers]]''' <!--- DNe (recommend Modern Algebra or MAT3013 --->
 
−
 
 
−
|| <!--Student Learning Outcomes-->
 
−
 
 
−
<!-- ------------------------------------------------------------ -->
 
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|-
 
−
 
 
−
 
 
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| <!--Date--> Week 2 
 
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−
|| <!--Sections-->
 
−
 
 
−
2.1
 
−
 
 
−
|| <!--Topics--> 
 
−
 
 
−
[[The Ordering Properties of the Real Numbers]]
 
−
 
 
−
|| <!--Prerequisite Skills-->
 
−
 
 
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* [[Solving Inequalities| Inequalities]] <!--- 1073- Mod R --->
 
−
* [[Algebraic properties of the Real Numbers]] <!--- 3213-2.1 --->
 
−
 
 
−
|| <!--Student Learning Outcomes-->
 
−
 
 
−
<!-- ------------------------------------------------------------ -->
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
| <!--Date--> Week 2   
 
−
 
 
−
|| <!--Sections-->
 
−
 
 
−
2.1
 
−
 
 
−
|| <!--Topics-->
 
−
 
 
−
[[Inequalities]]
 
−
 
 
−
|| <!--Prerequisite Skills-->
 
−
 
 
−
* [[The Ordering Properties of the Real Numbers]] <!--- 3213-2.1 --->
 
−
* [[The Algebraic Properties of the Real Numbers]] <!--- 3213-2.1 --->
 
−
 
 
−
|| <!--Student Learning Outcomes-->
 
−
 
 
−
<!-- ------------------------------------------------------------ -->
 
−
 
 
−
|-
 
−
 
 
−
 
 
−
| <!--Date--> Week 2/3 
 
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−
|| <!--Sections-->
 
−
 
 
−
2.2
 
−
 
 
−
|| <!--Topics--> 
 
−
 
 
−
[[Absolute Value and the Real Line]]
 
−
 
 
−
|| <!--Prerequisite Skills-->
 
−
 
 
−
* [[The Algebraic Properties of the Real Numbers]] <!--- 3213-2.1 --->
 
−
* [[Inequalities]] <!--- 3213-2.1 --->
 
−
 
 
−
|| <!--Student Learning Outcomes-->
 
−
 
 
−
<!-- ------------------------------------------------------------ -->
 
−
 
 
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|-
 
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| <!--Date--> Week 3  
 
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−
|| <!--Sections-->
 
−
 
 
−
2.3
 
  
 
|| <!--Topics-->  
 
|| <!--Topics-->  
  
−
[[Suprema, Infima, and the Completeness Property]]
+
[[RealNumbers:Completeness]]
 +
[[RealNumbers:Archimedean Property|Archimedean Property]]
 +
[[RealNumbers:Suprema of Subsets|Suprema of Subsets]]
 +
[[RealNumbers:Infima of Subsets|Infima of Subsets]]
 +
[[RealNumbers:Suprema of Subsets|Suprema of Subsets]]
 +
[[RealNumbers:Rational]]
 +
[[RealNumbers:Irrational]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[Inequalities]] <!--- 3213-2.1 --->
+
[[RealNumbers:Algebraic Properties]]
−
* [[Absolute Value and the Real Line]] <!--- 3213-2.2 --->
+
[[RealNumbers:Inequalities|Inequalities]]
 +
[[RealNumbers:Order]]
 +
[[RealNumbers:Absolute Value|Absolute Value]]
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
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|-
 
|-
  
−
 
+
| <!--Date--> Week 4  
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| <!--Date--> Week 3 
 
−
 
 
−
|| <!--Sections-->
 
−
 
 
−
2.4
 
−
 
 
−
|| <!--Topics--> 
 
−
 
 
−
[[Applications of the Supremum Property]]
 
−
 
 
−
|| <!--Prerequisite Skills-->
 
−
 
 
−
* [[Inequalities]] <!--- 3213-2.1 --->
 
−
* [[Absolute Value and the Real Line]] <!--- 3213-2.2 --->
 
−
* [[Suprema, Infima, and the Completeness Property]] <!--- 3213-2.3 --->
 
−
 
 
−
|| <!--Student Learning Outcomes-->
 
−
 
 
−
<!-- ------------------------------------------------------------ -->
 
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|-
 
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| <!--Date--> Week 3/4  
 
  
 
|| <!--Sections-->  
 
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|| <!--Topics-->   
 
|| <!--Topics-->   
  
−
[[Intervals]]
+
[[RealNumbers:Intervals|Intervals]]
 +
[[RealNumbers:Cardinality]]
 +
[[RealNumbers:Decimal Representation|Decimal Representation]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[Inequalities]] <!--- 3213-2.1 --->
+
[[RealNumbers:Completeness|Completeness]] <!--- 3213-2.1 --->
−
* [[Suprema, Infima, and the Completeness Property]] <!--- 3213-2.3 --->
+
[[RealNumbers:Archimedean Property|Archimedean Property]]
 +
[[RealNumbers:Bounded Subsets|Bounded Sets]]
 +
[[RealNumbers:Suprema of Subsets|Suprema of Subsets]]
 +
[[RealNumbers:Infima of Subsets|Infima of Subsets]]
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
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|-
 
|-
  
−
 
+
| <!--Date--> Week 5
−
| <!--Date--> Week 4
 
  
 
|| <!--Sections-->  
 
|| <!--Sections-->  
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|| <!--Topics-->   
 
|| <!--Topics-->   
  
−
[[Sequences and Their Limits]]
+
[[RealNumbers:Sequences|Sequences]]
 +
[[Sequences:Tails]]
 +
[[Sequences:Limits]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[Absolute Value and the Real Line]] <!--- 3213-1.1 --->
+
[[RealNumbers:Inequalities|Inequalities]]
−
* [[Countable Sets]] <!--- 3213-1.3 --->
+
[[RealNumbers:Absolute Value|Absolute Value]]
−
* [[Applications of the Supremum Property]] <!--- 3213-2.2 --->
 
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
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|-
 
|-
  
−
 
+
| <!--Date--> Week 6
−
| <!--Date--> Week 4 
 
  
 
|| <!--Sections-->  
 
|| <!--Sections-->  
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|| <!--Topics-->   
 
|| <!--Topics-->   
  
−
[[The Limit Laws for Sequences]]
+
[[Sequences:Limit Laws]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[Suprema, Infima, and the Completeness Property|Bounded sets]] <!--- 3213-2.3 --->
+
[[Sequences:Limits]]
−
* [[Sequences and Their Limits]] <!--- 3213-3.1 --->
 
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
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|-
 
|-
  
−
 
+
| <!--Date--> Week 7
−
| <!--Date--> Week 4/5   
 
  
 
|| <!--Sections-->  
 
|| <!--Sections-->  
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|| <!--Topics-->   
 
|| <!--Topics-->   
  
−
[[Monotone Sequences]]
+
[[Sequences:Monotone]]
 +
[[Euler's Number]]
  
 
|| <!--Prerequisite Skills-->  
 
|| <!--Prerequisite Skills-->  
  
−
* [[Mathematical Induction]] <!--- 3213-1.2 --->
+
[[Proofs:Induction]] <!--- 3213-1.2 --->
−
* [[Suprema, Infima, and the Completeness Property|Bounded sets]] <!--- 3213-2.3 --->
+
[[RealNumbers:Bounded Subsets|Bounded Sets]]
−
* [[The Limit Laws for Sequences|Bounded Sequences]] <!--- 3213-3.2 --->
+
[[Sequences:Limit Laws]]
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  
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* [[Cluster Point]] <!--- 3213-4.1 --->
 
* [[Cluster Point]] <!--- 3213-4.1 --->
−
* [[Intervals]] <!--- 3213-2.5 --->
+
* [[RealNumbers:Intervals|Intervals]] <!--- 3213-2.5 --->
  
 
|| <!--Student Learning Outcomes-->  
 
|| <!--Student Learning Outcomes-->  

Revision as of 14:09, 14 August 2020

The textbook for this course is Introduction to Real Analysis by Bartle and Sherbert

A comprehensive list of all undergraduate math courses at UTSA can be found here.

The Wikipedia summary of Real Analysis.

Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1

1.1

Sets:Definitions Sets:Operations Functions:Definition Functions:Domain Functions:Range Functions:Injective Functions:Bijective Functions:Composition Functions:Inverse Functions:ForwardImage Functions:InverseImage Functions:Restriction


Week 2

1.2

NaturalNumbers:Well Ordering Proofs:Induction Induction:Variants


Week 3

1.3


Finite and Infinite Sets Countable Sets

Sets:Operations NaturalNumbers:Well Ordering Proofs:Induction Functions:Injective Functions:Bijective


Week 3

2.1-2.2

Real Numbers RealNumbers:Algebraic Properties RealNumbers:Order Inequalities Absolute Value


Week 3

2.3-2.4

RealNumbers:Completeness Archimedean Property Suprema of Subsets Infima of Subsets Suprema of Subsets RealNumbers:Rational RealNumbers:Irrational

RealNumbers:Algebraic Properties Inequalities RealNumbers:Order Absolute Value


Week 4

2.5

Intervals RealNumbers:Cardinality Decimal Representation

Completeness Archimedean Property Bounded Sets Suprema of Subsets Infima of Subsets


Week 5

3.1

Sequences Sequences:Tails Sequences:Limits

Inequalities Absolute Value


Week 6

3.2

Sequences:Limit Laws

Sequences:Limits


Week 7

3.3

Sequences:Monotone Euler's Number

Proofs:Induction Bounded Sets Sequences:Limit Laws


Week 5

3.4

Subsequences


Week 5

3.4

The Bolzano Weierstrass Theorem


Week 5/6

3.4

The Limit Superior and Limit Inferior


Week 6

3.5

The Cauchy Criterion for Convergence


Week 6

3.6

Properly Divergent Sequences


Week 6/7

3.7

Introduction to Infinite Series


Week 12

4.1

Cluster Points


Week 12

4.1

The Definition of the Limit of a Function


Week 12/13

4.1

The Sequential Criterion and Divergence Criteria


Week 13

4.2

The Limit Theorems for Functions


Week 14

4.3

One Sided Limits


Week 14/15

4.3

Infinite Limits and Limits at Infinity


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