Math 240

Notes on Math 240: Introduction to Discrete Mathematics @University of Wisconsin-Madison Your comments and criticism are greatly welcomed.

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Kenneth H. Rosen, Discrete Mathematics and its Applications, seventh Edition

Lecture Notes

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0. Introductory Lecture

1. The Foundations: Logic and Proofs

1.1 Propositional Logic

1.2 Applications of Propositional Logic

1.3 Propositional Equivalences

1.4 Predicates and Quantifiers

1.5 Nested Quantifiers

1.6 Rules of Inference

1.7 Introduction to Proofs

1.8 Proof Methods and Strategy

2. Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

2.1 Sets

2.2 Set Operations

2.3 Functions

2.4 Sequences and Summations

2.5 Cardinality of Sets

2.6 Matrices

3. Algorithms

3.1 Algorithms

3.2 The Growth of Functions

3.3 Complexity of Algorithms

4. Number Theory and Cryptography

4.1 Divisibility and Modular Arithmetic

4.2 Integer Representations and Algorithms

4.3 Primes and Greatest Common Divisors

4.4 Solving Congruences

5. Induction and Recursion

5.1 Mathematical Induction

5.2 Strong Induction and Well-Ordering

5.3 Recursive Definitions and Structural Induction

5.4 Recursive Algorithms

6. Counting

6.1 The Basics of Counting

6.2 The Pigeonhole Principle

6.3 Permutations and Combinations

6.4 Binomial Coefficients and Identities

6.5 Generalized Permutations and Combinations

7. Discrete Probability

7.1 An Introduction to Discrete Probability

9. Relations

9.1 Relations and Their Properties

9.3 Representing Relations

9.5 Equivalence Relations

9.6 Partial Orderings

10. Graphs

10.1 Graphs and Graph Models

10.2 Graph Terminology and Special Types of Graphs

10.3 Representing Graphs and Graph Isomorphism

10.4 Connectivity

11. Trees

11.1 Introduction to Trees

Lecture Slides

Past Exams

Spring 2002 - Exam 1

Spring 2002 - Exam 2

Spring 2002 - Final

Spring 2005 - Exam 1

Spring 2005 - Exam 2

Spring 2005 - Final

Spring 2008 - Exam 1

Spring 2008 - Exam 2

Spring 2008 - Exam 3

Spring 2008 - Final 

Spring 2011 - Exam 1 (ExamSolution)

Spring 2011 - Exam 2 (Part I, Part II, Solution)

Spring 2011 - Final (ExamSolution)

Spring 2015 - Exam 2

Spring 2016 - Exam 2

Math 521

Notes on Math 521: Analysis I @ University of Wisconsin-Madison Your comments and criticism are greatly welcomed.

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Rudin, W. Principles of Mathematical Analysis. Third Edition

Lecture Notes

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Week 1
1/24 Number Systems, Irrationality of √2
1/26 Sets, Gaps in Q, Field
Week 2
1/29 Field, Order, Upper Bound and Lower Bound
1/31 Infimum and Supremum, Ordered Field
2/2 Ordered Field, Archimedean Property, Density of Q in R
Week 3
2/5 n-th Root of Real Number, Complex Numbers
2/7 Complex Numbers, Euclidean Spaces
2/9 Quiz
Week 4
2/12 Schwarz Inequality, Function, Cardinality
2/14 Finite and Infinite, Sequence
2/16 Set Operations, Countable and Uncountable
Week 5
2/19 Metric Space, Interval, Cell, Ball, Convex
2/21 Definitions in Metric Space
2/23 Neighborhood, Open and Closed, De Morgan's Law
Week 6
2/26 Open and Closed, Closure
2/28 Convergence and Divergence, Range, Bounded
3/2 Important Properties of Convergent Sequences
Week 7
3/5 Algebraic Limit Theorem
3/7 Convergence of Sequences in R^n, Compact Set
3/9 Exam 1
Week 8
3/12 Compact Subset, Cantor's Intersection Theorem
3/14 Nested Intervals Theorem, Compactness of k-cell
3/16 Heine-Borel, Weierstrass, Subsequence
Week 9
3/19 Cauchy Sequence, Diameter
3/21 Cauchy Sequence, Complete Metric Space, Monotonic
3/23 Upper and Lower Limits
Week 10
4/2 Some Special Sequences
4/4 Series, Cauchy Criterion for Series, Comparison Test
4/6 Convergence Tests for Series
Week 11
4/9 Power Series, Absolute Convergence, Rearrangement
4/11 Rearrangement, Limit of Functions
4/13 Exam 2
Week 12
4/16 Continuous Function and Open Set
4/18 Continuity and Compactness, Extreme Value Theorem
4/20 Uniform Continuity and Compactness
Week 13
4/23 Connected Set, Intermediate Value Theorem
4/25 Derivative, Chain Rule, Local Extrema
4/27 Mean Value Theorem, Monotonicity, Taylor's Theorem
Week 14
4/30 Riemann-Stieltjes Integral, Refinement
5/2 Fundamental Theorem of Calculus
5/4 Sequence of Functions, Uniform Convergence

Math 541

Notes on Math 541: Modern Algebra @ University of Wisconsin-Madison Your comments and criticism are greatly welcomed.

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Abstract Algebra, by Dummit and Foote, Third Edition, 2004

Lecture Notes

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Week 1
1/24 Divides, Equivalence Relations
1/26 Well-ordering of Z
Week 2
1/29 Division Algorithm, gcd
1/31 Euclidean Algorithm
2/2 Equivalence Class, Z/nZ, Group
Week 3
2/5 Group, Well-definedness, Z/nZ
2/7 (Z/nZ)*, Properties of Group
2/9 Order, Symmetric Group
Week 4
2/12 Symmetric Group, Cycle
2/14 Homomorphism, Isomorphism
2/16 Order, Homomorphism, Subgroup
Week 5
2/19 Dihedral Groups, Subgroup
2/21 Cyclic Group, lcm, Order of g^a
2/23 Cyclic Subgroup, Generating Set of a Group
Week 6
2/26 Finitely Generated Group
2/28 Coset, Normal Subgroup
3/2 Exam 1
Week 7
3/5 Quotient Group, Index, Lagrange's Theorem
3/7 Corollaries of Lagrange's Theorem
3/9 The First & Second Isomorphism Theorems
Week 8
3/12 The Third Isomorphism Theorem
3/14 Transposition, Sign of Permutation
3/16 Homework 6, The Correspondence Theorem
Week 9
3/19 Sign of Permutation, Alternating Group
3/21 Subgroups of A_4, Group Action, Orbit, Stabilizer
3/23 Orbit, Stabilizer, Cayley's Theorem
Week 10
4/2 Conjugacy Class, The Class Equation
4/4 Cauchy's Theorem, Recognizing Direct Products
4/6 Homework 8, Properties of Finite Abelian Group
Week 11
4/9 Fundamental Theorem of Finite Abelian Groups
4/11 Definition of Ring
4/13 Exam 2
Week 12
4/16 Properties of Ring, Zero-Divisor, Unit
4/18 Field, Product Ring, Integral Domain
4/20 Product Ring, Finite Domain and Field, Subring
Week 13
4/23 Polynomial Ring, Ideal, Principal Ideal
4/25 Examples of Ideals, Quotient Ring
4/27 Isomorphism Theorems for Rings
Week 14
4/30 Generating Ideal, Maximal Ideal, Prime Ideal
5/2 Prime Ideal, Euclidean Domain
5/4 Review, Galois Theory

抽象代数 Abstract Algebra


Table of Contents

第1讲 集合的定义

第2讲 集合的运算

第3讲 集合间的关系

第4讲 映射

第5讲 罗素悖论(选修)

第6讲 势与基数(选修)

第7讲 定义良好

第8讲 群的定义

第9讲 子群与生成

第10讲 循环群与阶

第11讲 陪集与指数

第12讲 拉格朗日定理

第13讲 共轭与正规子群

第14讲 商群

第15讲 同态与同构

第16讲 群同构定理

第17讲 群作用

第18讲 合成列(选修)

第19讲 自由群(选修)

第20讲 稳定子,中心化, 正规化子

第21讲 类等式定理

第22讲 n次对称群

第23讲 自同构

第24讲 Z/n

第25讲 半直积

第26讲 西罗定理(选修)

第27讲 西罗定理的应用(选修)

第28讲 可解群

CS/ECE 252

CS/ECE 252: Introduction to Computer Engineering Source:
Universal Computing Devices MP4 Flash
Trends and Complexity MP4 Flash
Abstraction MP4 Flash
Electrical Information MP4 Flash
Number Representation MP4 Flash
Base Conversion MP4 Flash
Counting MP4 Flash
Arithmetic MP4 Flash
Signed Numbers MP4 Flash
Sign Extension and Overflow MP4 Flash
Fixed- and Floating-Point MP4 Flash
Logic Functions MP4 Flash
Combinational Logic MP4 Flash
Combinational Building Blocks MP4 Flash
Sequential Logic and Flip-Flops MP4 Flash
Finite State Machines MP4 Flash
Registers and Memory MP4 Flash
Basic Processor Model MP4 Flash
Instructions and the Instruction Cycle MP4 Flash
LC-3 ISA and Processor Overview MP4 Flash
LC-3 Operate Instructions MP4 Flash
Simple Programs with LC-3 Operate Instructions MP4 Flash
Basic LC-3 Data Movement MP4 Flash
More LC-3 Data Movement MP4 Flash
LC-3 Control Flow MP4 Flash
Programming Techniques MP4 Flash
LC-3 Assembly Language MP4 Flash
LC-3 Memory Allocation MP4 Flash
LC-3 Assembler MP4 Flash
Subroutines MP4 Flash
Programming With Subroutines MP4 Flash
I/O Concepts MP4 Flash
LC-3 I/O MP4 Flash
Operating Systems MP4 Flash
LC-3 TRAPs MP4 Flash