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## Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

Mathematical logic and proof, mathematical induction, counting methods, recurrence relations, algorithms and complexity, graph theory and graph algorithms.

This course is equivalent to MATH 2602.

*Discrete Mathematics with Graph Theory*, Goodaire and Parmenter, 3rd edition

Logic and proofs: Compound statements, proofs, truth tables, sets, relations, functions.

Algorithms and recursion. Division algorithm, Euclidean algorithm, congruence, mathematical induction, recursively defined sequences, recurrence relations and the characteristic polynomial, algorithms, complexity, searching and sorting.

Combinatorics: Inclusion/exclusion principle, addition and multiplication rules, pigeonhole principle, permutations, combinations, repetitions, probability, binomial theorem.

Graph Theory: Isomorphism, Eulerian paths, Hamiltonian paths, Dijkstra's algorithm, trees, Kruskal's algorithm, planar graphs, chromatic number.

## CMPS/MATH 2170: Discrete MathematicsSpring 2017

This course is an introduction to the area of Discrete Mathematics. The word "discrete" should be understood in the sense that the mathematical objects which we will be studying are not continuous. This is an extremely broad area within mathematics and we will only provide an introduction to a select few topics (listed below). All of the topics which we shall cover are fundamental for both computer science and mathematics.

### Synopsis:

- Logic and Proofs
- Naive Set Theory
- Mathematical Induction and Recursion
- Combinatorics
- Relations
- Graph Theory

### Instructor:

Dr. Vladimir Zamdzhiev

Office hours: WF 11:00am - 12:00 noon, Stanley Thomas 314 and by appointment

Email: vzamdzhi

### Teaching Assistant:

Selcuk Karakoc

Office hours: T 1:00pm - 2:00pm, Gibson Hall 305 and by appointment

E-mail: skarakoc

### Time & Place:

**Lecture:**MWF 02:00pm - 02:50pm, Stanley Thomas 302**Lab:**R 12:30pm - 01:45pm Gibson 325

### Grading:

Grades will be posted on Canvas. The weighted average will determine your letter grade as follows:

A >= 90% B >= 80% C >= 70% D >= 60% F < 60%

### Exams and quizzes:

There will be 12 quizzes in total. Quizzes will take place during the labs. The two lowest scoring quizzes will be dropped when determining your grade. No make-up quizzes are allowed.

In addition, there will also be a midterm and a final exam. The final exam will cover all topics of the course. The midterm exam will cover only the material which we have discussed in class prior to it.

Missing any exam or quiz will result in a grade of zero for it. A request for a make-up exam must be given to the instructor prior to the exam date (documentation may be required).

All exams (including quizzes) will be **closed book**.

The midterm exam is scheduled for **Thursday, March 9, 2017, 12:30pm - 01:45pm, Gibson 325**.

The final exam is scheduled for **Friday, May 5, 2017, 08:00am - 10:30am, Stanley Thomas 302**.

### Homework:

There will be homework assignments almost every week. Assignments are due at the beginning of the lab session the week after they are posted. Some homework assignments will have optional (more difficult) problems which can be solved to improve your overall grade. Homework assignments will be posted on the course webpage one week before they are due. The solutions to the homework assignments must be your own work. Handing in late homework is not allowed, except if you have a valid reason which must be communicated to the instructor before the due date of the assignment.

## Introduction to Discrete Mathematics &ndash MATH 250

Logic and proofs, set theory, Boolean algebra, functions, sequences, matrices, algorithms, modular arithmetic, mathematical induction and combinatorics.

For information regarding prerequisites for this course, please refer to the Academic Course Catalog.

### Rationale

Discrete mathematics, the study of finite mathematical systems, provides students with mathematical ideas, notations and skills which are critical to, for example, formulating what an algorithm is supposed to achieve, proving if it meets the specification, and analyzing its time and space complexity. Discrete mathematics is essential to the study of computer science.

### Measurable Learning Outcomes

Upon successful completion of this course, the student will be able to:

- Construct valid mathematical arguments using logical connectives and quantifiers.
- Verify the correctness of a mathematical argument using symbolic logic and truth tables.
- Construct a proof using direct proof, proof by contradiction, and proof by cases.
- Perform operations on discrete structures such as sets, discrete functions, relations, sequences, and matrices.
- Analyze algorithms, determine algorithmic complexity, and apply algorithms to solve problems.
- Express a Boolean function as a Boolean sum of Boolean products of the variables and their complements.
- Use Boolean algebra to model the circuitry of electronic devices.
- Use relations to solve problems involving communications networks, project scheduling.

### Course Assignment

**Textbook readings and video presentations**

**Course Requirements Checklist**

After reading the Course Syllabus and Student Expectations, the student will complete the related checklist found in Module/Week 1.

**Learn Smart Reading Assignments (8)**

The student will complete reading assignments within the ConnectMath software associated with the textbook.

The student will complete handwritten homework assignments and submit them in Blackboard each week.

Each quiz will cover the Reading & Study material for the assigned modules/weeks. Each quiz will be open-book/open-notes, have a 1 hour time limit, and be completed in ConnectMath software.

The student will complete exams during Modules/Weeks 2, 4, 6, and 8. Each exam will be open-book/open-notes, cover 2 modules/weeks of material, and have a 2 hour time limit. All tests are handwritten.

## Introduction to Discrete Mathematics &ndash MATH 250

Logic and proofs, set theory, Boolean algebra, functions, sequences, matrices, algorithms, modular arithmetic, mathematical induction and combinatorics.

For information regarding prerequisites for this course, please refer to the Academic Course Catalog.

### Rationale

Discrete mathematics, the study of finite mathematical systems, provides students with mathematical ideas, notations and skills which are critical to, for example, formulating what an algorithm is supposed to achieve, proving if it meets the specification, and analyzing its time and space complexity. Discrete mathematics is essential to the study of computer science.

### Measurable Learning Outcomes

Upon successful completion of this course, the student will be able to:

- Construct valid mathematical arguments using logical connectives and quantifiers.
- Verify the correctness of a mathematical argument using symbolic logic and truth tables.
- Construct a proof using direct proof, proof by contradiction, and proof by cases.
- Perform operations on discrete structures such as sets, discrete functions, relations, sequences, and matrices.
- Analyze algorithms, determine algorithmic complexity, and apply algorithms to solve problems.
- Express a Boolean function as a Boolean sum of Boolean products of the variables and their complements.
- Use Boolean algebra to model the circuitry of electronic devices.
- Use relations to solve problems involving communications networks, project scheduling.

### Course Assignment

**Textbook readings and video presentations**

**Course Requirements Checklist**

After reading the Course Syllabus and Student Expectations, the student will complete the related checklist found in Module/Week 1.

**Learn Smart Reading Assignments (8)**

The student will complete reading assignments within the ConnectMath software associated with the textbook.

The student will complete handwritten homework assignments and submit them in Blackboard each week.

Each quiz will cover the Reading & Study material for the assigned modules/weeks. Each quiz will be open-book/open-notes, have a 1 hour time limit, and be completed in ConnectMath software.

The student will complete exams during Modules/Weeks 2, 4, 6, and 8. Each exam will be open-book/open-notes, cover 2 modules/weeks of material, and have a 2 hour time limit. All tests are handwritten.

## 1: Introduction to Discrete Mathematics - Mathematics

packer

Course Information:

The primary aim of this course is to introduce the ideas of rigor and proof in mathematics through the study of topics in discrete mathematics. Topics will include the nature of mathematics (definitions, theorems, proofs, and counterexamples), basic logic, lists and sets, counting and relations, permutations and symmetry, some basic number theory and cryptography, and basic graph theory.

The material to be covered will be contained in the book "Mathematics: A Discrete Introduction" by Edward Scheinerman, third edition. We will cover most of Chapters 1, 2, 3 and 4, and parts of Chapters 5, 6, 7, and a small portion of Chapter 9. We will occasionally refer to other textbooks.

Prerequisite:

MATH 1300 (Calculus 1) or the equivalent.

Course Text:

We will use the text "Mathematics: A Dicrete Introduction" by Edward Scheinerman, Third Edition.

Course Syllabus: For course syllabus, click here!

Please note this course also has a website on Canvas, with links to lecture notes, resources, details about Zoom, Netiquette, Proctorio software, COVID-19 regulations, etc.

- Homework will be assigned every week. Some, but not all, of the problems will be graded. The assessment of homework performance will count for 20% of the final grade. Your lowest two lowest homework scores will be dropped. No late homework is accepted. You will use Canvas to upload your homework. Copying solutions from Chegg.com or any other homework-solution-providing website for your solutions in your homework is a violation of the Honor Code and violations will be reported to the Honor Council.
- In-class worksheets or projects in small breakout rooms will be done for part of the class every Friday. Participation in these activities will count 5% towards your final grade. You will use Canvas to access and/or upload the worksheets.
- There are two in-class midterms, the first on Wednesday, Feb. 24, 2021, 12:40 p.m.- 1:40 p.m., and Friday April 2, 2021, 12:40 p.m.- 1:40 p.m., each of which counts 20% towards your final grade. You will need Proctorio software to take these exams.
- There will be two (longer) written projects, focusing on communication of mathematics and mathematical proofs. These will count 5% each. You will use Canvas to upload your written projects.
- In-class comprehensive final exam - Monday May 3, 2021, 1:30 p.m. - 4 p.m., in-class: 25% of final grade. You will need Proctorio software to take this exam.

Lecture Hours and Venue:

MWF 12:40 p.m.-1:30 p.m. remotely via ZOOM.

Office Hours:

MWF 11 a.m.- 12 noon via ZOOM, and by appointment.

- Assignment 1, due Friday, Jan. 22, 2021, uploaded to Canvas by 11:59 p.m.
- Assignment 2: Read Chapter 1.5, 1.6, 1,7 Section 3, p. 7: #6, #8, #12 Section 4, p. 13-14: #1, #3, #4, #5, #6, plus two extra problems on Canvas due Friday Jan. 29, 2021, 11:59 p.m. uploaded to Canvas.
- Assignment 3: Read 1.7 again, read Chapter 2 Section 8 and Section 9. Writtten HW: pp. 22-23: #5.5, #5.14, #5.18, #23 pp. 24-35: #6.5, #6.9, #6.10, #6.13 pp. .28-29: #73, #7.7, #7.10 (a), (c), #7.13(a) p 38: #8.6, plus the extra problem on Canvas: due Friday Feb. 5, 2021, 11:59 p.m. uploaded to Canvas.
- Assignment 4: Re-read Section 9, and read Sections 10, 11, 12 of the textbook. Fo problems in Section 8, pp. 38-39: #8.6, #8.7, #8.12 (a), (c), (e),(f) Section 9, pp. 42-43: #9.4, #9.5, #9.6, #9.8 (a) (d), #9 Section 10, pp.50-51:#10.1 (a), (c), (f), #10.3 (a), (c), (e), #10.6 (a), (b), (d), plus extra problem on Canvas: due Friday Feb. 12, 2021, 11:50 p.m. uploaded to Canvas.
- Assignment 5, due Monday Feb. 22, 2021
- Assignment 6, due Monday, March 8, 2021
- Assignment 7, due Monday, March 15, 2021
- Assignment 8, due Monday, March 22, 2021
- Assignment 9, due Wednesday, March 31, 2021
- Assignment 10, due Wednesday, April 14, 2021.
- Assignment 11, due Wednesday, April 21, 2021.
- Assignment 12, due Wednesday, April 28, 2021.

- Solutions to Midterm 1, given Friday, February 24, 2021.
- Solutions to Midterm 2, given Friday, April 2, 2021.
- Sample final exam from May 2015.

- Writing assigment 1, final peer-reviewed draft due uploaded to Canvas on Friday, Feb. 12, 2021, 11:59 p.m. uploaded to Canvas.
- Writing assigment 2.

- D. Bernoulli
- G. Cantor
- L. Euler
- Pierre Fermat
- C.F. Gauss
- I. Newton
- B. Pascal
- G. F. von Leibniz

Back to the home page of Judith A. Packer

Last modified April 30, 2015.

## Contents

The history of discrete mathematics has involved a number of challenging problems which have focused attention within areas of the field. In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, but not proved until 1976 (by Kenneth Appel and Wolfgang Haken, using substantial computer assistance). [10]

In logic, the second problem on David Hilbert's list of open problems presented in 1900 was to prove that the axioms of arithmetic are consistent. Gödel's second incompleteness theorem, proved in 1931, showed that this was not possible – at least not within arithmetic itself. Hilbert's tenth problem was to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. In 1970, Yuri Matiyasevich proved that this could not be done.

The need to break German codes in World War II led to advances in cryptography and theoretical computer science, with the first programmable digital electronic computer being developed at England's Bletchley Park with the guidance of Alan Turing and his seminal work, On Computable Numbers. [11] At the same time, military requirements motivated advances in operations research. The Cold War meant that cryptography remained important, with fundamental advances such as public-key cryptography being developed in the following decades. Operations research remained important as a tool in business and project management, with the critical path method being developed in the 1950s. The telecommunication industry has also motivated advances in discrete mathematics, particularly in graph theory and information theory. Formal verification of statements in logic has been necessary for software development of safety-critical systems, and advances in automated theorem proving have been driven by this need.

Computational geometry has been an important part of the computer graphics incorporated into modern video games and computer-aided design tools.

Several fields of discrete mathematics, particularly theoretical computer science, graph theory, and combinatorics, are important in addressing the challenging bioinformatics problems associated with understanding the tree of life. [12]

Currently, one of the most famous open problems in theoretical computer science is the P = NP problem, which involves the relationship between the complexity classes P and NP. The Clay Mathematics Institute has offered a $1 million USD prize for the first correct proof, along with prizes for six other mathematical problems. [13]

### Theoretical computer science Edit

Theoretical computer science includes areas of discrete mathematics relevant to computing. It draws heavily on graph theory and mathematical logic. Included within theoretical computer science is the study of algorithms and data structures. Computability studies what can be computed in principle, and has close ties to logic, while complexity studies the time, space, and other resources taken by computations. Automata theory and formal language theory are closely related to computability. Petri nets and process algebras are used to model computer systems, and methods from discrete mathematics are used in analyzing VLSI electronic circuits. Computational geometry applies algorithms to geometrical problems, while computer image analysis applies them to representations of images. Theoretical computer science also includes the study of various continuous computational topics.

### Information theory Edit

Information theory involves the quantification of information. Closely related is coding theory which is used to design efficient and reliable data transmission and storage methods. Information theory also includes continuous topics such as: analog signals, analog coding, analog encryption.

### Logic Edit

Logic is the study of the principles of valid reasoning and inference, as well as of consistency, soundness, and completeness. For example, in most systems of logic (but not in intuitionistic logic) Peirce's law (((*P*→*Q*)→*P*)→*P*) is a theorem. For classical logic, it can be easily verified with a truth table. The study of mathematical proof is particularly important in logic, and has applications to automated theorem proving and formal verification of software.

Logical formulas are discrete structures, as are proofs, which form finite trees [14] or, more generally, directed acyclic graph structures [15] [16] (with each inference step combining one or more premise branches to give a single conclusion). The truth values of logical formulas usually form a finite set, generally restricted to two values: *true* and *false*, but logic can also be continuous-valued, e.g., fuzzy logic. Concepts such as infinite proof trees or infinite derivation trees have also been studied, [17] e.g. infinitary logic.

### Set theory Edit

Set theory is the branch of mathematics that studies sets, which are collections of objects, such as

In discrete mathematics, countable sets (including finite sets) are the main focus. The beginning of set theory as a branch of mathematics is usually marked by Georg Cantor's work distinguishing between different kinds of infinite set, motivated by the study of trigonometric series, and further development of the theory of infinite sets is outside the scope of discrete mathematics. Indeed, contemporary work in descriptive set theory makes extensive use of traditional continuous mathematics.

### Combinatorics Edit

Combinatorics studies the way in which discrete structures can be combined or arranged. Enumerative combinatorics concentrates on counting the number of certain combinatorial objects - e.g. the twelvefold way provides a unified framework for counting permutations, combinations and partitions. Analytic combinatorics concerns the enumeration (i.e., determining the number) of combinatorial structures using tools from complex analysis and probability theory. In contrast with enumerative combinatorics which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Design theory is a study of combinatorial designs, which are collections of subsets with certain intersection properties. Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, special functions and orthogonal polynomials. Originally a part of number theory and analysis, partition theory is now considered a part of combinatorics or an independent field. Order theory is the study of partially ordered sets, both finite and infinite.

### Graph theory Edit

Graph theory, the study of graphs and networks, is often considered part of combinatorics, but has grown large enough and distinct enough, with its own kind of problems, to be regarded as a subject in its own right. [18] Graphs are one of the prime objects of study in discrete mathematics. They are among the most ubiquitous models of both natural and human-made structures. They can model many types of relations and process dynamics in physical, biological and social systems. In computer science, they can represent networks of communication, data organization, computational devices, the flow of computation, etc. In mathematics, they are useful in geometry and certain parts of topology, e.g. knot theory. Algebraic graph theory has close links with group theory. There are also continuous graphs however, for the most part, research in graph theory falls within the domain of discrete mathematics.

### Probability Edit

Discrete probability theory deals with events that occur in countable sample spaces. For example, count observations such as the numbers of birds in flocks comprise only natural number values <0, 1, 2, . >. On the other hand, continuous observations such as the weights of birds comprise real number values and would typically be modeled by a continuous probability distribution such as the normal. Discrete probability distributions can be used to approximate continuous ones and vice versa. For highly constrained situations such as throwing dice or experiments with decks of cards, calculating the probability of events is basically enumerative combinatorics.

### Number theory Edit

Number theory is concerned with the properties of numbers in general, particularly integers. It has applications to cryptography and cryptanalysis, particularly with regard to modular arithmetic, diophantine equations, linear and quadratic congruences, prime numbers and primality testing. Other discrete aspects of number theory include geometry of numbers. In analytic number theory, techniques from continuous mathematics are also used. Topics that go beyond discrete objects include transcendental numbers, diophantine approximation, p-adic analysis and function fields.

### Algebraic structures Edit

Algebraic structures occur as both discrete examples and continuous examples. Discrete algebras include: boolean algebra used in logic gates and programming relational algebra used in databases discrete and finite versions of groups, rings and fields are important in algebraic coding theory discrete semigroups and monoids appear in the theory of formal languages.

### Calculus of finite differences, discrete calculus or discrete analysis Edit

A function defined on an interval of the integers is usually called a sequence. A sequence could be a finite sequence from a data source or an infinite sequence from a discrete dynamical system. Such a discrete function could be defined explicitly by a list (if its domain is finite), or by a formula for its general term, or it could be given implicitly by a recurrence relation or difference equation. Difference equations are similar to differential equations, but replace differentiation by taking the difference between adjacent terms they can be used to approximate differential equations or (more often) studied in their own right. Many questions and methods concerning differential equations have counterparts for difference equations. For instance, where there are integral transforms in harmonic analysis for studying continuous functions or analogue signals, there are discrete transforms for discrete functions or digital signals. As well as the discrete metric there are more general discrete or finite metric spaces and finite topological spaces.

### Geometry Edit

Discrete geometry and combinatorial geometry are about combinatorial properties of *discrete collections* of geometrical objects. A long-standing topic in discrete geometry is tiling of the plane. Computational geometry applies algorithms to geometrical problems.

### Topology Edit

Although topology is the field of mathematics that formalizes and generalizes the intuitive notion of "continuous deformation" of objects, it gives rise to many discrete topics this can be attributed in part to the focus on topological invariants, which themselves usually take discrete values. See combinatorial topology, topological graph theory, topological combinatorics, computational topology, discrete topological space, finite topological space, topology (chemistry).

### Operations research Edit

Operations research provides techniques for solving practical problems in engineering, business, and other fields — problems such as allocating resources to maximize profit, and scheduling project activities to minimize risk. Operations research techniques include linear programming and other areas of optimization, queuing theory, scheduling theory, and network theory. Operations research also includes continuous topics such as continuous-time Markov process, continuous-time martingales, process optimization, and continuous and hybrid control theory.

### Game theory, decision theory, utility theory, social choice theory Edit

Cooperate | Defect | |

Cooperate | −1, −1 | −10, 0 |

Defect | 0, −10 | −5, −5 |

Payoff matrix for the Prisoner's dilemma, a common example in game theory. One player chooses a row, the other a column the resulting pair gives their payoffs |

Decision theory is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision.

Utility theory is about measures of the relative economic satisfaction from, or desirability of, consumption of various goods and services.

Social choice theory is about voting. A more puzzle-based approach to voting is ballot theory.

Game theory deals with situations where success depends on the choices of others, which makes choosing the best course of action more complex. There are even continuous games, see differential game. Topics include auction theory and fair division.

### Discretization Edit

Discretization concerns the process of transferring continuous models and equations into discrete counterparts, often for the purposes of making calculations easier by using approximations. Numerical analysis provides an important example.

### Discrete analogues of continuous mathematics Edit

In applied mathematics, discrete modelling is the discrete analogue of continuous modelling. In discrete modelling, discrete formulae are fit to data. A common method in this form of modelling is to use recurrence relation.

In algebraic geometry, the concept of a curve can be extended to discrete geometries by taking the spectra of polynomial rings over finite fields to be models of the affine spaces over that field, and letting subvarieties or spectra of other rings provide the curves that lie in that space. Although the space in which the curves appear has a finite number of points, the curves are not so much sets of points as analogues of curves in continuous settings. For example, every point of the form V ( x − c ) ⊂ Spec K [ x ] = A 1

### Hybrid discrete and continuous mathematics Edit

The time scale calculus is a unification of the theory of difference equations with that of differential equations, which has applications to fields requiring simultaneous modelling of discrete and continuous data. Another way of modeling such a situation is the notion of hybrid dynamical systems.

## 1: Introduction to Discrete Mathematics - Mathematics

MWF 9:00-9:50 Henry 140

Text: Kenneth H. Rosen: Discrete Mathematics and Its Applications, 6th Edition, McGraw-Hill

Instructor: Elizabeth Csima

Office Hours: Wednesdays 4:00 - 5:20, Illini Hall 332. Also by appointment.

You can check your grades here.

More information about your grades here.

There will not be a quiz in class on Wednesday May 4th. Wednesday's class will serve as review to prepare for the final. The final exam takes place on Wednesday May 11th, 8:00am-11:00am in Henry 140 (our classroom). It covers all of the material taught during the semester.

There will be review session on Monday May 9th at 5pm-6pm, in Altgeld 241.

Please check your grades on the score reporting system. If you see any discrepancies please let me know via email.

Homework #1 due Friday January 28th

Section 2.1: 8, 28(a)(d)

Section 2.2: 4, 14, 18*

Section 2.3: 2**, 12**, 14**, 18**, 32

*Please provide proofs for your answers, similar to examples done during lecture and example 10 of Section 2.2.

**For these problems if the answer is "no" give specific example to justify your answer. If the answer is "yes" no further justification is required.

Homework #2 Due Friday February 4th

Section 4.1: 6, 12, 14, 18, 20, 32, 40, 48

Homework #3 Due Friday February 11th

Section 5.1*: 8, 12, 20, 30, 34, 38, 42

Section 5.2**: 4, 14, 16

*Most of the the questions in this section have numerical answers. To receive full credit you must show how you arrived at your answer (i.e. what calculation are you making? What rules are you applying?). For instance if you are using product rule and arrive at 120 = 6*5*4 as an answer it is not enough to just write "120" for your solution on your homework. You should explain what 6, 5 and 4 are counting and why the product rule applies in your situation.

**Most of the problems in this section use the pigeonhole principle and have numerical answers. In your solutions make sure to explain how you are using the pigeonhole principle to arrive at your answer.

Suggested exercises from Section 5.3: 9, 10, 13, 17, 19, 33

Homework #4 Due Friday February 25th

Section 5.4: 8, 12, 22(a)*,(b)

Section 5.5: 10, 12, 16, 22, 30

*Hint: Count the number of ways to pick a k-element subset of an r-element subset of an n-element set in two different ways.

Homework #5 Due Friday March 4th

Section 6.1: 6, 12, 16, 30

Section 6.2: 16, 18*, 26, 30

For the problems on this homework please specify what counting methods you are using when completing the problems.

*18(c) requires the use of a calculator.

Suggested exercises:

Section 6.3: 1, 3, 5, 9, 11

Section 7.1: 1, 7, 9, 23, 25, 27

Section 7.2: 3

Homework #6 Due Friday March 18th

Section 7.2: 24, 26, 28, 32

Section 7.5: 6, 8, 10, 16

Homework #7 Due Friday April 1st

Section 7.6: 4, 8, 10, 14

Section 8.1*: 6(a)-(f), 8

Section 8.5: 16, 30(a)(b), 36, 40, 44(a)(b)(c)

*To earn full credit justify all of your assertions by either giving a short proof to show the given property holds or a specific counterexample to show why a property fails.

Homework #8 Due Friday April 8th

Section 9.1: 10, 12, 16

Section 9.2*: 2, 4, 22, 26

*questions 50 and 52 have been dropped from the this week's assignment.

Suggested Exercises

Section 9.3: 1, 3, 5, 7, 11, 15, 17, 35, 39, 41

Homework #9 Due Friday April 22nd

Section 9.4: 6, 12, 18, 20

Section 9.5: 2, 4, 10, 28, 30, 32, 34, 36

Homework #10 Due Friday April 29th

Section 9.6: 2, 4, 6(c)(d), 12(a)(b), 14

Section 9.7: 4, 6, 8, 12, 16, 20, 22, 24

Section 9.8: 2, 4, 8, 10, 18, 24

## Mathematics (MATH)

This Learning Support course provides corequisite support in mathematics for students enrolled in MATH 1111. Topics will parallel topics being studied in MATH 1111 and the essential quantitative skills needed to be successful.

**MATH 1111. College Algebra. 4 Credit Hours.**

This course is symbolically intensive, functional approach to algebra that incorporates the use of appropriate technology. Emphasis will be placed on the study of functions and their graphs, inequalities, and linear, quadratic, piece-wise defined, rational, polynomial, exponential, and logarithmic functions. Appropriate applications will be included.

**MATH 1113. Precalculus. 4 Credit Hours.**

Analytic geometry, the function concept, polynomials, exponential, logarithms, trigonometric functions, mathematical induction, and the theory of equations. May only be used for degree credit with departmental approval.

**MATH 11X3. Transfer Precalculus. 3 Credit Hours.**

**MATH 1501. Calculus I. 4 Credit Hours.**

Differential calculus and basic integral calculus including the fundamental theorem of calculus. Credit not allowed for both MATH 1501 and 1712.

**MATH 1503. Calculus I for the Life Sciences. 4 Credit Hours.**

Differential and basic calculus: sequences, difference equations, limits, continuity, differentiation, integration, applications. The topics parallel those of MATH 1501 with applications from life sciences.

**MATH 1504. Calculus I for the Life Sciences. 4 Credit Hours.**

Taylor approximations, introduction to differential equations, linear algebra, and introduction to multivariable calculus. Motivating examples drawn from life sciences.

**MATH 1512. Honors Calculus II. 4 Credit Hours.**

The topics covered parallel those of 1502 with a somewhat more intensive and rigorous treatment. Credit not allowed for both honors calculus and the corresponding regular calculus course. Credit not allowed for both MATH 1512 and MATH 1522. Credit not allowed for both MATH 1512 and MATH 15X2.

**MATH 1550. Introduction to Differential Calculus. 3 Credit Hours.**

An introduction to differential calculus including applications and the underlying theory of limits for functions and sequences. Credit not awarded for both MATH 1550 and MATH 1501, MATH 1551, or MATH 1503.

**MATH 1551. Differential Calculus. 2 Credit Hours.**

Differential calculus including applications and the underlying theory of limits for functions and sequences. Credit not awarded for both MATH 1551 and MATH 1501, MATH 1503, or MATH 1550.

**MATH 1552. Integral Calculus. 4 Credit Hours.**

Integral calculus: Definite and indefinite integrals, techniques of integration, improper integrals, infinite series, applications. Credit not awarded for both MATH 1552 and MATH 1502, MATH 1504, MATH 1512 or MATH 1555.

**MATH 1553. Introduction to Linear Algebra. 2 Credit Hours.**

An introduction to linear alegbra including eigenvalues and eigenvectors, applications to linear systems, least squares. Credit not awarded for both MATH 1553 and MATH 1522, MATH 1502, MATH 1504, MATH 1512, MATH 1554 or MATH 1564.

**MATH 1554. Linear Algebra. 4 Credit Hours.**

Linear algebra eigenvalues, eigenvectors, applications to linear systems, least squares, diagnolization, quadratic forms.

**MATH 1555. Calculus for Life Sciences. 4 Credit Hours.**

Overview of intergral calculus, multivariable calculus, and differential equations for biological sciences. Credit not awarded for both MATH 1555 and MATH 1552, MATH 1502, MATH 1504, MATH 1512 or MATH 2550.

**MATH 1564. Linear Algebra with Abstract Vector Spaces. 4 Credit Hours.**

This is an intensive first course in linear algebra including the theories of linear transformations and abstract vector spaces. Credit not awarded for both MATH 1564 and MATH 1553, MATH 1554, MATH 1522, MATH 1502, MATH 1504 or MATH 1512.

**MATH 15X1. Transfer Calculus I. 3 Credit Hours.**

**MATH 15X2. Transfer Calculus II. 3,4 Credit Hours.**

This course includes the treatment of single variable calculus in MATH 1502. This course is not equivalent to MATH 1502. Credit not allowed for both MATH 15X2 and MATH 1502. Credit not allowed for both MATH 15X2 and MATH 1512.

**MATH 1601. Introduction to Higher Mathematics. 3 Credit Hours.**

This course is designed to teach problem solving and proof writing. Mathematical subject matter is drawn from elementary number theory and geometry.

**MATH 1711. Finite Mathematics. 4 Credit Hours.**

Linear equations, matrices, linear programming, sets and counting, probability and statistics.

**MATH 1712. Survey of Calculus. 4 Credit Hours.**

Techniques of differentiation, integration, application of integration to probability and statistics, multidimensional calculus. Credit not allowed for both MATH 1712 and 1501.

**MATH 17X1. Transfer Finite Math. 3 Credit Hours.**

**MATH 17X2. Transfer Survey-Calc. 3 Credit Hours.**

**MATH 1803. Special Topics. 3 Credit Hours.**

Courses on special topics of current interest in Mathematics.

**MATH 1X51. Transfer Differential Calc. 2,3 Credit Hours.**

**MATH 1X52. Transfer Integral Calculus. 3,4 Credit Hours.**

**MATH 1X53. Transfer Intro Linear Algebra. 2,3 Credit Hours.**

**MATH 1X54. Transfer Linear Algebra. 2,3 Credit Hours.**

**MATH 1X55. Transfer Calculus for Life Sci. 2,3 Credit Hours.**

**MATH 1XXX. Mathematics Elective. 1-21 Credit Hours.**

**MATH 2106. Foundations of Mathematical Proof. 3 Credit Hours.**

An introduction to proofs in advanced mathematics, intended as a transition to upper division courses including Abstract Algebra I and Analysis I.

**MATH 2406. Abstract Vector Spaces. 3 Credit Hours.**

A proof-based development of linear algebra and vector spaces, with additional topics such as multilinear algebra and group theory.

**MATH 24X1. Transfer Calculus III. 3 Credit Hours.**

**MATH 24X3. Transfer Diff Equations. 3 Credit Hours.**

**MATH 2550. Introduction to Multivariable Calculus. 2 Credit Hours.**

Vectors in three dimensions, curves in space, functions of several variables, partial derivatives, optimization, integration of functions of several variables. Vector Calculus not covered. Credit will not be awarded for both MATH 2550 and MATH 2605 or MATH 2401 or MATH 2551 or MATH 1555.

**MATH 2551. Multivariable Calculus. 4 Credit Hours.**

Multivariable calculus: Linear approximation and Taylor's theorems, Lagrange multiples and constrained optimization, multiple integration and vector analysis including the theorems of Green, Gauss, and Stokes. Credit will not be awarded for both MATH 2551 and MATH 2401 or MATH 2411 or MATH 2561.

**MATH 2552. Differential Equations. 4 Credit Hours.**

Methods for obtaining numerical and analytic solutions of elementary differential equations. Applications are also discussed with an emphasis on modeling. Credit not awarded for both MATH 2552 and MATH 2403 or MATH 2413 or MATH 2562.

**MATH 2561. Honors Multivariable Calculus. 4 Credit Hours.**

The topics covered parallel those of MATH 2551 with a somewhat more intensive and rigorous treatment. Credit not awarded for both MATH 2561 and MATH 2401 or MATH 2411 or MATH 2551.

**MATH 2562. Honors Differential Equations. 4 Credit Hours.**

The topics covered parallel those of MATH 2552 with a somewhat more intensive and rigorous treatment.

**MATH 2603. Introduction to Discrete Mathematics. 4 Credit Hours.**

Mathematical logic and proof, mathematical induction, counting methods, recurrence relations, algorithms and complexity, graph theory and graph algorithms. Credit not awarded for both MATH 2603 and MATH 2602.

**MATH 2605. Calculus III for Computer Science. 4 Credit Hours.**

Topics in linear algebra and multivariate calculus and their applications in optimization and numerical methods, including curve fitting, interpolation, and numerical differentiation and integration.

**MATH 2698. Undergraduate Research Assistantship. 1-12 Credit Hours.**

Independent research conducted under the guidance of a faculty member.

**MATH 2699. Undergraduate Research. 1-12 Credit Hours.**

Independent research conducted under the guidance of a faculty member.

**MATH 26X2. Transfer Linear & Disc Math. 3 Credit Hours.**

**MATH 26X3. Transfer Discrete Math. 3 Credit Hours.**

**MATH 2801. Special Topics. 1 Credit Hour.**

Courses on special topics of current interest in mathematics.

**MATH 2802. Special Topics. 2 Credit Hours.**

Courses on special topics of current interest in mathematics.

**MATH 2803. Special Topics. 3 Credit Hours.**

Courses on special topics of current interest in mathematics.

**MATH 2804. Special Topics. 4 Credit Hours.**

Courses on special topics of current interest in mathematics.

**MATH 2805. Special Topics. 5 Credit Hours.**

Courses on special topics of current interest in mathematics.

**MATH 2X51. Transfer Multivariable Calc. 3,4 Credit Hours.**

**MATH 2X52. Transfer Differential Equation. 3,4 Credit Hours.**

**MATH 2XXX. Mathematics Elective. 1-21 Credit Hours.**

**MATH 3012. Applied Combinatorics. 3 Credit Hours.**

Elementary combinatorial techniques used in discrete problem solving: counting methods, solving linear recurrences, graph and network models, related algorithms, and combinatorial designs.

**MATH 3022. Honors Applied Combinatorics. 3 Credit Hours.**

Topics are parallel to those of MATH 3012 with a more rigorous and intensive treatment. Credit is not allowed for both MATH 3012 and 3022.

**MATH 3215. Introduction to Probability and Statistics. 3 Credit Hours.**

This course is a problem-oriented introduction to the basic concepts of probability and statistics, providing a foundation for applications and further study.

**MATH 3225. Honors Probability and Statistics. 3 Credit Hours.**

The topics covered parallel those of MATH 3215, with a more rigorous and intensive treatment. Credit is not allowed for both MATH 3215 and 3225.

**MATH 3235. Probability Theory. 3 Credit Hours.**

This course is a mathematical introduction to probability theory, covering random variables, moments, multivariable distributions, law of large numbers, central limit theorem, and large deviations. Credit not awarded for both MATH 3235 and MATH 3215 or 3225 or 3670.

**MATH 3236. Statistical Theory. 3 Credit Hours.**

An introduction to theoretical statistics for students with a background in probability. A mathematical formalism for inference on experimental data will be developed. Credit not awared for both MATH 3236 and MATH 3215 or 3225 or 3670.

**MATH 3406. A Second Course in Linear Algebra. 3 Credit Hours.**

This course will cover important topics in linear algebra not usually discussed in a first-semester course, featuring a mixture of theory and applications.

**MATH 3670. Probability and Statistics with Applications. 3 Credit Hours.**

Introduction to probability, probability distributions, point estimation, confidence intervals, hypothesis testing, linear regression and analysis of variance. Students cannot receive credit for both MATH 3670 and MATH 3770 or ISYE 3770 or CEE 3770.

**MATH 3801. Special Topics. 1 Credit Hour.**

Courses on special topics of current interest in mathematics.

**MATH 3802. Special Topics. 2 Credit Hours.**

Courses on special topics of current interest in mathematics.

**MATH 3803. Special Topics. 3 Credit Hours.**

Courses on special topics of current interest in mathematics.

**MATH 3804. Special Topics. 4 Credit Hours.**

Courses on special topics of current interest in mathematics.

**MATH 3805. Special Topics. 5 Credit Hours.**

Courses on special topics of current interest in mathematics.

**MATH 3XXX. Mathematics Elective. 1-21 Credit Hours.**

**MATH 4012. Algebraic Structures in Coding Theory. 3 Credit Hours.**

Introduction to linear error correcting codes with an emphasis on the algebraic tools required, including matrices vector spaces, groups, polynomial rings, and finite fields.

**MATH 4022. Introduction to Graph Theory. 3 Credit Hours.**

The fundamentals of graph theory: trees, connectivity, Euler torus, Hamilton cycles, matchings, colorings, and Ramsey theory.

**MATH 4032. Combinatorial Analysis. 3 Credit Hours.**

Combinatorial problem-solving techniques including the use of generating functions, recurrence relations, Polya theory, combinatorial designs, Ramsey theory, matroids, and asymptotic analysis.

**MATH 4080. Senior Project I. 2 Credit Hours.**

The first of a two-course sequence of faculty-directed independent research culminating in the writing of a senior thesis and its presentation.

**MATH 4090. Senior Project II. 2 Credit Hours.**

The second course of a two-course sequence of faculty-directed independent research culminating in the writing of a senior thesis and its presentation.

**MATH 4107. Introduction to Abstract Algebra I. 3 Credit Hours.**

This course develops in the theme of "Arithmetic congruence and abstract algebraic structures". Strong emphasis on theory and proofs.

**MATH 4108. Introduction to Abstract Algebra II. 3 Credit Hours.**

Continuation of Abstract Algebra I, with emphasis on Galois theory, modules, polynomial fields, and the theory of linear associative algebra.

**MATH 4150. Introduction to Number Theory. 3 Credit Hours.**

Primes and unique factorization, congruences, Chinese remainder theorem, Diophantine equations, Diophantine approximations, quadratic reciprocity. Applications such as fast multiplication, factorization, and encryption.

**MATH 4221. Probability with Applications I. 3 Credit Hours.**

Simple random walk and the theory of discrete time Markov chains.

**MATH 4222. Probability with Applications II. 3 Credit Hours.**

Renewal theory, Poisson processes and continuous time Markov processes, including an introduction to Brownian motion and martingales.

**MATH 4255. Monte Carlo Methods. 3 Credit Hours.**

Probability distributions, limit laws, and applications through the computer.

**MATH 4261. Mathematical Statistics I. 3 Credit Hours.**

Sampling distributions, Normal, t, chi-square, and f distributions. Moment-generating function methods, Bayesian estimation, and introduction to hypothesis testing.

**MATH 4262. Mathematical Statistics II. 3 Credit Hours.**

Hypothesis testing, likelihood ratio tests, nonparametric tests, bivariate and multivariate normal distributions.

**MATH 4280. Introduction to Information Theory. 3 Credit Hours.**

The measurement and quantification of information. These ideas are applied to the probabilistic analysis of the transmission of information over a channel along which random distortion of the message occurs.

**MATH 4305. Topics in Linear Algebra. 3 Credit Hours.**

Finite dimensional vector spaces, inner product spaces, least squares, linear transformations, the spectral theorem for normal transformations. Applications to convex sets, positive matrices, difference equations.

**MATH 4317. Analysis I. 3 Credit Hours.**

Real numbers, topology of Euclidean spaces, Cauchy sequences, completeness, continuity and compactness, uniform continuity, series of functions, Fourier series.

**MATH 4318. Analysis II. 3 Credit Hours.**

Differentiation of functions of one real variable, Riemann-Stieltjes integral, the derivative in Rn, and integration in Rn.

**MATH 4320. Complex Analysis. 3 Credit Hours.**

Topics from complex function theory, including contour integration and conformal mapping.

**MATH 4347. Partial Differential Equations I. 3 Credit Hours.**

Method of characteristics for first- and second-order partial differential equations, conservation laws and shocks, classification of second-order systems and applications.

**MATH 4348. Partial Differential Equations II. 3 Credit Hours.**

Green's functions and fundamental solutions. Potential, diffusion, and wave equations.

**MATH 4431. Introductory Topology. 3 Credit Hours.**

Point set topology, topological spaces and metric spaces, continuity and compactness, homotopy, and covering spaces.

**MATH 4432. Introduction to Algebraic Topology. 3 Credit Hours.**

Introduction to algebraic methods in topology. Includes homotopy, the fundamental group, covering spaces, simplicial complexes. Applications to fixed point theory and group theory.

**MATH 4441. Differential Geometry. 3 Credit Hours.**

The theory of curves, surfaces, and more generally, manifolds. Curvature, parallel transport, covariant differentiation, Gauss-Bonet theorem.

**MATH 4541. Dynamics and Bifurcations I. 3 Credit Hours.**

A broad introduction to the local and global behavior of nonlinear dynamical systems arising from maps and ordinary differential equations.

**MATH 4542. Dynamics and Bifurcations II. 3 Credit Hours.**

A continuation of Dynamics and Bifurcations I.

**MATH 4580. Linear Programming. 3 Credit Hours.**

A study of linear programming problems, including the simplex method, duality, and sensitivity analysis with applications to matrix games, interger programming, and networks.

**MATH 4581. Classical Mathematical Methods in Engineering. 3 Credit Hours.**

The Laplace transform and applications, Fourier series, boundary value problems for partial differential equations.

**MATH 4640. Numerical Analysis I. 3 Credit Hours.**

Introduction to numerical algorithms for some basic problems in computational mathematics. Discussion of both implementation issues and error analysis.

**MATH 4641. Numerical Analysis II. 3 Credit Hours.**

Introduction to the numerical solution of initial and boundary value problems in differential equations.

**MATH 4695. Undergraduate Internship. 1-21 Credit Hours.**

Undergraduate internship for academic credit.

**MATH 4698. Undergraduate Research Assistantship. 1-12 Credit Hours.**

Independent research conducted under the guidance of a faculty member.

**MATH 4699. Undergraduate Research. 1-12 Credit Hours.**

Independent research conducted under the guidance of a faculty member.

**MATH 4755. Mathematical Biology. 3 Credit Hours.**

Problems from the life sciences and the mathematical methods for solving them are presented. The underlying biological and mathematical principles and the interrelationships are emphasized. Crosslisted with BIOL 4755.

**MATH 4777. Vector and Parallel Scientific Computation. 3 Credit Hours.**

Scientific computational algorithms on vector and parallel computers. Speed-up and algorithm complexity, interprocesses communication, synchronization, modern algorithms for linear systems, programming techniques, code optimization. Crosslisted with CS 4777.

**MATH 4782. Quantum Information and Quantum Computing. 3 Credit Hours.**

Introduction to quantum computing and quantum information theory, formalism of quantum mechanics, quantum gates, algorithms, measurements, coding, and information. Physical realizations and experiments. Crosslisted with PHYS 4782.

**MATH 4801. Special Topics. 1 Credit Hour.**

Courses on special topics of current interest in mathematics.

**MATH 4802. Special Topics. 2 Credit Hours.**

Courses on special topics of current interest in mathematics.

**MATH 4803. Special Topics. 3 Credit Hours.**

Courses on special topics of current interest in mathematics.

**MATH 4804. Special Topics. 4 Credit Hours.**

Courses on special topics of current interest in mathematics.

**MATH 4805. Special Topics. 5 Credit Hours.**

Courses on special topics of current interest in mathematics.

**MATH 4873. Special Topics. 3 Credit Hours.**

This course enables the school of Mathematics to comply with requests for courses in selected topics.

**MATH 4999. Reading or Research. 1-21 Credit Hours.**

Reading or research in topics of current interest.

**MATH 4XXX. Mathematics Elective. 1-21 Credit Hours.**

**Georgia Institute of Technology**

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