## Math 240

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

Homepage

Syllabus

# Textbook

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

# Lecture Notes

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

# 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.

Homepage

Syllabus

# Textbook

Rudin, W. Principles of Mathematical Analysis. Third Edition

# Lecture Notes

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.

Homepage

Syllabus

# Textbook

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

# Lecture Notes

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

## 1.3 Propositional Equivalences ## 1.2 Applications of Propositional Logic 