Fall 2011 Introduction to Modern Mathematics

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Mathematics 311 - Fall 2011

Introduction to Modern Mathematics

Course Description

Introduction to logic, basic properties of sets, relations, functions, one-to-one functions, setequivalence, Cantor's Theorem, countable and uncountable sets. Prerequisite: MTH 234.

For a more detailed course description, go tohttp://www2.sfasu.edu/math/courses/syllabi/MTH312Syllabus.pdf

Course Time and Meeting Place

MTH 311 meets in Math 357 at 11-11:50 MWF.

Instructor

Thomas W. Judson, Associate Professor Department of Mathematics and StatisticsOffice: Math 316TEL: (936) 468-1704Email: judsontw @ sfasu.eduOffice Hours:

Monday: 10-10:50 AM (in Math 316)Tuesday: 9:30-10:30 AM (in Math 316) and 6-7:15 PM (in the AARC)Wednesday: 10-10:50 AM (in Math 316) and 6-7:15 PM (in the AARC)Thursday: 9:30-10:30 AM (in Math 316)Friday: 10-10:50 AM (in Math 316)By appointment

Course Goals

To be able to demonstrate comprehension of core mathematical concepts such asaxiomatic systems, mathematical proof, and logical argument.To be able to execute mathematical procedures accurately, appropriately, andefficiently.To be able to apply the principles of logic to develop and analyze conjectures andproofs.To be able to demonstrate competence in using various mathematical tools,including technology, to formulate, represent, and solve problems.

Learning Objectives

Upon successfully completing Math 311, you should have acquired a solid foundation of thefollowing topics and be able to move directly into more advanced courses.

Read and interpret written mathematics and communicate their reasoning both orallyand in written form.Translate between symbolic logic notation and standard English.Understand and interpret compound statements, logical arguments, and fallacies.Make appropriate inferences based on conditional and biconditional statements.Understand the role of quantifiers in mathematical statements.Formulate reasonable conjectures and construct rigorous, well written proofs using avariety of proof techniques (including direct and indirect proofs).

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Examinations



Construct appropriate counterexamples to disprove statements.Understand the principle of mathematical induction and use it in the formulation ofmathematical proofs.State and use important definitions in set theory.Understand and construct proofs concerning subsets and set equality.Recognize and prove theorems about equivalence relations, including congruencerelations on the set of integers.Understand the notion of function and be able to state and use definitions of one-to-one, onto, image and pre-image.Understand the notions of infinite set and cardinality and use them to prove thatgiven sets have the same cardinality.Understand a proof of the uncountability of the set of real numbers.Apply their understanding of logic and proof in an appropriate mathematical contextwhich may include number theory, graph theory, topology, analysis, algebra or otherrelevant topics.

Textbook

The textbook for this course is

Carol Schumacher. Chapter Zero: Fundamental Notions of Abstract Mathematics,second edition. Addison Wesley, Boston, 2000.

ISBN-10: 0201437244ISBN-13: 978-0201437249

Grading and Exams

There will be three exams and a final exam. Exams may be in class or take home.Your course grade will be determined as follows:

Component Percentage

Class Work (Homework, Class Presentations, Participation) 45%

Exam I (In Class) 10%

Exam II (Take Home) 10%

Exam III (Take Home) 10%

Final Exam (Take Home and In Class) 25%

Semester numerical scores will be converted into letter grades according to thefollowing method.

Range of numerical values Corresponding Letter

90-100 A

80-89 B

70-79 C



60-69 D

0-59 F

When we calculate your final grade at the end of the course, we will calculate ascore on a 0-100 point scale using the scores that you have obtained during thecourse, and using the grade breakdown given above. Your course grade will then beobtained using this table. In the event of a fractional score, we will always round upto the nearest integer.

Resurrection Policy. If you score better on the final exam than your lowest midtermexam, we will replace your midterm grade with your final exam grade. Theressurection policy does not apply to your homework grade or to the arithmeticexam.

Exam Policy

Exams are scheduled far in advance, and it is impossible to move the time or date.However, in rare cases where it is impossible for an individual to take the exam atthe scheduled time, we will work with you to make other arrangements. Exceptionsfor taking the exam out of sequence are the following:

1. A medical excuse. Please provide proper documentation according touniversity rules.

2. A University sponsored event such as an athletic tournament, a play, or amusical performance. Your coach or director must contact us in advance.Athletic practices and rehearsals do not fall into this category.

3. A religious holiday. Please send a short email explaining the situation.4. Extreme hardship such as a family emergency. Please have the proper

university office us.The above are the only allowable excuses for taking the exam before the scheduledtime. Under no circumstances do we give late exams. Since we can onlyaccomodate a limited number of students taking the exam at an earlier time, pleasemake sure that you fall into one of the above categories before you contact us. If youmiss an exam due to illness or a family emergency, you will not be penalized. Wewill assign you a grade based on the rest of your coursework. If you have a conflictwith the final exam, you must contact the Registrar. Only the Registrar can schedulean out-of-sequence final exam.

Homework and Classroom Presentations

There is no question that the best way to learn math is by doing math, andhomework exercises are an essential part of any math course. If you just go to amath class and watch the teacher work problems, but do not actually try doing anyproblems on your own, then there is very little chance you will really learn thesubject. It is also very unlikely that you will do well on exams without workingthrough homework problems ahead of time. When doing homework, do not just writedown answers. Think about the problems posed, your strategies, the meaning ofyour computations, and the answers you get. The main point is not to come up withspecific answers to the specific problems you are working on, but to develop anunderstanding of what you are doing so that you can apply your reasoning to a widerange of similar situations. It is very unlikely that later on in life you will see exactlythe same math problems you are working on now, so learn the material in such away that you are prepared to use your general knowledge of mathematics in thefuture, not just how to apply particular formulas for very specific problems.



You are encouraged to form study groups with other students in the class so thatyou can discuss your work with each other; however, all work submitted must bewritten up individually. Make sure that even if you do work in groups, that you comeaway with the ability to explain everything you end up writing up in your homework.

During the course of the semester, you will be expected to present some of yoursolutions to assigned homework problems to the rest of the class. This will be part ofyour grade, so it is very important that you come to class prepared. You will beasked to write up all of your assignments in a homework notebook. Generally, yourhomework notebook will be due on Friday and will be returned to you by followingMonday's class. Since getting behind in a math class is one of the mostuncomfortable things you can do to yourself, homework must be turned in on time.Since we will drop your lowest homework grade, we will not accept late homeworknotebooks. For the complete homework grading policy, go to the assignments pageof the Math 311 website.

The Classroom

Any questions you ask in class will likely be ones that other students will wantanswered as well, so get over any hesitation you might have and ask questions asthe material is presented. You will not be penalized for doing this, no matter howtrivial or simple you think your questions might seem. Remember, the class is beingheld for you to learn the material, not just to give you a time to copy notes off of ablackboard, so be sure to get help when you need it and stay involved in your class.

Getting Help with Math 311

Remember to take advantage of office hours. You can also try just dropping by myoffice. If I am not preoccupied, I will be happy to answer any questions that youmight have.

Add/Drop Policy

The Add/Drop Policy can be found at http://www.sfasu.edu/policies/add_drop.asp

Attendance Policy

Regular attendance is expected in Math 311. Attendance and Excused AbsencesPolicy can be found athttp://www.sfasu.edu/policies/class_attendance_excused_abs.asp Be sure to look atthe exams policy on the exams page of the Math 311 website.

Withheld Grades Semester Grades Policy (A-54)

Ordinarily, at the discretion of the instructor of record and with the approval of theacademic chair/director, a grade of WH will be assigned only if the student cannotcomplete the course work because of unavoidable circumstances. Students mustcomplete the work within one calendar year from the end of the semester in whichthey receive a WH, or the grade automatically becomes an F. If students register forthe same course in future terms the WH will automatically become an F and will becounted as a repeated course for the purpose of computing the grade point average.

The circumstances precipitating the request must have occurred after the last day inwhich a student could withdraw from a course. Students requesting a WH must bepassing the course with a minimum projected grade of C.



Students with Disabilities

To obtain disability related accommodations, alternate formats and/or auxiliary aids,students with disabilities must contact the Office of Disability Services (ODS),Human Services Building, and Room 325, 468-3004 / 468-1004 (TDD) as early aspossible in the semester. Once verified, ODS will notify the course instructor andoutline the accommodation and/or auxiliary aids to be provided. Failure to requestservices in a timely manner may delay your accommodations. For additionalinformation, go to http://www.sfasu.edu/disabilityservices/

Acceptable Student Behavior

Classroom behavior should not interfere with the instructor's ability to conduct theclass or the ability of other students to learn from the instructional program (see theStudent Conduct Code, policy D-34.1). Unacceptable or disruptive behavior will notbe tolerated. Students who disrupt the learning environment may be asked to leaveclass and may be subject to judicial, academic or other penalties. This prohibitionapplies to all instructional forums, including electronic, classroom, labs, discussiongroups, field trips, etc. The instructor shall have full discretion over what behavior isappropriate/inappropriate in the classroom. Students who do not attend classregularly or who perform poorly on class projects/exams may be referred to the EarlyAlert Program. This program provides students with recommendations for resourcesor other assistance that is available to help SFA students succeed.

Please be respectful of your fellow students and your instructor. Cell phone use andtexting are not allowed in class. Remember to turn your cell phone off or place it inquiet mode before entering the classroom.

Academic Integrity (A-9.1)

Academic integrity is a responsibility of all university faculty and students. Facultymembers promote academic integrity in multiple ways including instruction on thecomponents of academic honesty, as well as abiding by university policy onpenalties for cheating and plagiarism.

Definition of Academic Dishonesty. Academic dishonesty includes both cheating andplagiarism. Cheating includes but is not limited to (1) using or attempting to useunauthorized materials to aid in achieving a better grade on a component of a class;(2) the falsification or invention of any information, including citations, on anassigned exercise; and/or (3) helping or attempting to help another in an act ofcheating or plagiarism. Plagiarism is presenting the words or ideas of anotherperson as if they were your own. Examples of plagiarism are (1) submitting anassignment as if it were one's own work when, in fact, it is at least partly the work ofanother; (2) submitting a work that has been purchased or otherwise obtained froman Internet source or another source; and (3) incorporating the words or ideas of anauthor into one's paper without giving the author due credit.

Any acts of academic dishonesty will be dealt with according to University policy.Penalties for academic dishonesty may result in a failing grade for the assignment,failing the course, or even dismissal from the university.

Please read the complete Academic Integrity Policy athttp://www.sfasu.edu/policies/academic_integrity.asp


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Mathematics 311


Exam Policy

Exams are scheduled far in advance, and it is impossible to move the time or date. However,in rare cases where it is impossible for an individual to take the exam at the scheduled time,we will work with you to make other arrangements. Exceptions for taking the exam out ofsequence are the following:

1. A medical excuse. Please provide proper documentation according to university rules.2. A University sponsored event such as an athletic tournament, a play, or a musical

performance. Your coach or director must contact us in advance. Athletic practicesand rehearsals do not fall into this category.

3. A religious holiday. Please send a short email explaining the situation.4. Extreme hardship such as a family emergency. Please have the proper university

office contact us.

The above are the only allowable excuses for taking the exam before the scheduled time.Under no circumstances do we give late exams. Since we can only accomodate a limitednumber of students taking the exam at an earlier time, please make sure that you fall intoone of the above categories before you contact us. If you miss an exam due to illness or afamily emergency, you will not be penalized. We will assign you a grade based on the rest ofyour coursework. If you have a conflict with the final exam, you must contact the Registrar.Only the Registrar can schedule an out-of-sequence final exam.

Fall 2011Home

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Last modified: August 5, 2011


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Mathematics 311


Schedule of Homework Assignments

The schedule of homework assignments can be found on the Calender page.

Homework Policy

One of the goals of this course is for you to learn to think and communicate mathematically.This means that homework assignments should be written with justification andexplanations of your steps in English where appropriate. See the course textbook andnotes for examples of well-written solutions. Since many of the exam problems ask forjustification, this will be good practice.

Please submit your homework notebook on time. We cannot accept homework submittedby email. Since we will drop your lowest homework grade, please do not ask us to acceptlate homework assignments.

Homework Grading

More elaborate problems or parts of a multi-part problem will be worth three points. Wegrade more involved homework problems according to the following rubric.

4 Points. Solution is exemplary. Four points will only be awarded for solutions toparticularly difficult problems or very elegant solutions. Generally, 3 points will be themost that you can earn for a solution.3 Points. Work is completely accurate and essentially perfect. Work is thoroughlydeveloped, neat, and easy to read. Complete sentences are used whereappropriate.2 Points. Work is good, but incompletely developed, hard to read, unexplained, orjumbled. Answers that are not explained may received 2 points even if correct. Thework contains the right idea but is flawed.1 Points. Work is sketchy. There is some correct work, but most of the work isincorrect.0 Points. Work is minimal or non-existent. Solution is completely incorrect.

Making Your Homework Easy to Read and Easy to Grade

Make sure your handwriting is legible.Problems should be clearly labeled and numbered on the left side of the page.There should also be a visible separation between problems.You should leave the entire left margin blank so that the grader can use this spacefor scoring and comments.To ensure that each problem is graded, problems and solutions should be written inthe order that they are assigned.It is good practice to first work out the solutions to homework problems on scratchpaper, and then to neatly write up your solutions. This will help you turn in a cleanfinished product.You should write up your solutions by yourself. You should always acknowledge anyhelp received at the top of the assignment or in the right-hand margin.

Presentation Grading

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Mathematics 311


Calendar

Week 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17

Week Date TopicHomework

(Exercises marked with a "*" should be writtenup in your homework notebook.)

1 Monday 8/29/11

CourseOrientation § 1.1. True orFalse?

Read Chapter 0 in Chapter Zero: FundamentalNotions of Abstract Mathematics

Wednesday8/31/11

§ 1.2. Statementsand Predicates. § 1.3.Quantification. § 1.4.MathematicalStatements.

Exercises 1.2.1, 1.2.2, 1.3.1*, 1.3.2*, 1.3.3*, 1.4.3.

Friday 9/2/11

§ 1.5.MathematicalImplication. § 1.6. CompoundStatements andTruth Tables

Exercises 1.5.2*, 1.5.3, 1.6.2, 1.6.4*.

2 Monday 9/5/11

Labor Day - NoClass

Wednesday9/7/11

§ 1.7. Learningfrom TruthTables.

Exercises 1.7.1*, 1.7.3*, 1.7.5, 1.7.7*, 1.7.8*.

Friday 9/9/11

§ 1.8. NegatingStatements.

Exercises 1.8.1, 1.8.2, 1.8.3*, 1.8.4, 1.8.5*, 1.8.7,1.8.8*, 1.8.9, 1.8.10*, 1.8.11*.

3 Monday 9/12/11

§ 1.9. ExistenceTheorems § 1.10.UniquenessTheorems. § 1.11. ExamplesandCounterexamples.§ 1.12. DirectProof.

Exercises 1.11.1, 1.11.2*, 1.12.1*.

Wednesday9/14/11

§ 1.13. Proof byContrapositive. § 1.14. Proof byContradiction. § 1.15. ProvingTheorems: What

Exercises 1.13.1*, 1.13.3*, 1.13.4*, 1.14.1*.1.14.2, 1.14.4*.

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Now? Friday

9/16/11§ 2.1. Sets andSet Notation. § 2.2. Subsets. § 2.3. SetOperations.

Exercises 2.2.2*, 2.2.3, 2.2.4*, 2.2.6, 2.3.2, 2.3.3,2.3.5, 2.3.7, 2.3.8*, 2.3.10, 2.3.14, 2.3.15*.

4 Monday 9/19/11

§ 2.3. SetOperations.

Wednesday9/21/11

§ 2.4. TheAlgebra of Sets.

Exercises 2.4.1, 2.4.2, 2.4.3, 2.4.4*, 2.4.5*, 2.4.7,2.4.8*, 2.4.9*.

Friday 9/23/11

§ 2.4. TheAlgebra of Sets.

5 Monday 9/26/11

§ 2.5. The PowerSet.

Exercises 2.5.2, 2.5.3, 2.5.4*, 2.5.5*, 2.5.6, 2.5.7*,2.6.1.

Wednesday9/28/11

§ 2.5. The PowerSet. § 2.6. Russell'sParadox.

Exercises 2.6.1.

Friday 9/30/11

Exam I

6 Monday 10/3/11

§ 3.1.MathematicalInduction. § 3.2. UsingInduction.

Exercises 3.1.1, 3.2.2*, 3.2.3*, 3.2.4*, 3.2.5*,3.2.6*.

Wednesday10/5/11

§ 3.2. UsingInduction.

Friday 10/7/11

§ 3.2. UsingInduction. § 3.3. CompleteInduction.

Exercises 3.3.2*, 3.3.3*.

7 Monday 10/10/11

§ 4.1. Relations. Exercises 4.1.3, 4.1.6, 4.1.9, 4.1.10*.

Wednesday10/12/11

§ 4.2. Orderings. Exercises 4.2.4, 4.2.6*, 4.2.7, 4.2.8, 4.2.9, 4.2.12,4.2.13, 4.2.14*, 4.2.15*, 4.2.17, 4.2.18*, 4.2.20*,4.2.21, 4.2.22*, 4.2.25*, 4.2.26*.

Friday 10/14/11

§ 4.2. Orderings.

8 Monday 10/17/11

§ 4.3.EquivalenceRelations.

Exercises 4.3.2, 4.3.4, 4.3.5, 4.3.7*, 4.3.8*,4.3.10, 4.3.11*, 4.3.12, 4.3.14, 4.3.15, 4.3.16*,4.3.17*, 4.3.19, 4.3.20*, 4.3.21*, 4.3.23.

Wednesday10/19/11

§ 4.3.EquivalenceRelations.

Friday 10/21/11

§ 4.4. Graphs. Exercises 4.4.2, 4.4.5, 4.4.6*, 4.4.8, 4.4.9*,4.4.11, 4.4.12*, 4.4.13, 4.4.14*, 4.4.15*, 4.4.16*,4.4.19, 4.4.20, 4.4.21, 4.4.22, 4.4.25, 4.4.26,4.4.28*, 4.4.29*, 4.4.30*, 4.4.31*.

9 Monday 10/24/11

§ 4.4. Graphs.



Wednesday10/26/11

§ 4.4. Graphs.

Friday 10/28/11

Exam II Due(Take homeexam) § 5.1. BasicIdeas.

Exercises 5.1.2, 5.1.3, 5.1.4, 5.1.6, 5.1.7*, 5.1.9,5.1.10, 5.1.11, 5.1.12, 5.1.13*, 5.1.14, 5.1.15,5.1.16*.

10 Monday 10/31/11

§ 5.2.Composition andInverses.

Exercises 5.2.2, 5.2.3*, 5.2.4, 5.2.5, 5.2.6*, 5.2.7*,5.2.9*, 5.2.10*.

Wednesday11/2/11

§ 5.3. Images andInverse Images.

Exercises 5.3.3, 5.3.4, 5.3.5*, 5.3.6*, 5.3.8, 5.3.9,5.3.10, 5.3.11*, 5.3.12*.

Friday 11/4/11

§ 5.5. Sequences. Exercises 5.5.2, 5.5.4*, 5.5.7, 5.5.8*, 5.5.9*,5.5.11, 5.5.13, 5.5.14, 5.5.15*, 5.5.16, 5.5.19,5.5.20, 5.5.21*, 5.5.22*, 5.5.24*, 5.5.25*, 5.5.26*,5.5.27, 5.5.28*.

11 Monday 11/7/11

§ 5.5. Sequences.§ 5.6. BinaryOperations.

Exercises 5.6.3, 5.6.4, 5.6.6, 5.6.8, 5.6.9, 5.6.10*,5.6.11.

Wednesday11/9/11

§ 6.1. NaturalNumbers andIntegers.

Exercises 6.1.2, 6.1.3*, 6.1.4*.

Friday 11/11/11

§ 6.2. Divisibilityin the Integers.

Exercises 6.2.1*, 6.2.2, 6.2.3, 6.2.5, 6.2.6*, 6.2.8*,6.2.9*, 6.2.10, 6.2.12, 6.2.13*, 6.2.14*, 6.2.15,6.2.16*, 6.2.17*, 6.2.18, 6.2.19, 6.2.20.

12 Monday 11/14/11

§ 6.4. RelativelyPrime Integers.

Exercises 6.4.1*, 6.4.2*, 6.4.3*, 6.4.4*.

Wednesday11/16/11

§ 6.6.CongruenceModulo n.

Exercises 6.6.1*, 6.6.3, 6.6.4*, 6.6.5, 6.6.6*.

Friday 11/18/11

§ 6.7. DivisibilityModulo n.

Exercises 6.7.1*, 6.7.2, 6.7.3, 6.7.4, 6.7.5*, 6.7.6*,6.7.7*.

13 Monday 11/21/11

§ 7.1. Galileo'sParadox.

Exercises 7.1.2, 7.1.3*, 7.1.5*, 7.1.6, 7.1.8, 7.1.9.

Wednesday11/23/11

ThanksgivingRecess - Noclass

Friday 11/25/11

ThanksgivingRecess - Noclass

14 Monday 11/28/11

§ 7.2. InfiniteSets.

Exercises 7.2.3*, 7.2.4, 7.2.5*, 7.2.6*, 7.2.7*.

Wednesday11/30/11

§ 7.3. CountableSets.

Exercises 7.3.3*, 7.3.4*, 7.3.5*, 7.3.6*, 7.3.7*,7.3.9, 7.3.10*, 7.3.11.

Friday 12/2/11

Exam III Due(Take homeexam) § 7.4. BeyondCountability.

Exercises 7.4.1, 7.4.3*, 7.4.4*, 7.4.5*, 7.4.6*,7.4.8*,

15 Monday § 7.5. Comparing Exercises 7.5.2*, 7.5.4*, 7.5.5*, 7.5.6*, 7.5.7,



12/5/11 Cardinalities. 7.5.8. Wednesday

12/7/11§ 7.5. ComparingCardinalities.

Friday 12/9/11

sect& 7.6. TheContinuumHypothesis.

Exercise 7.6.1*.

16 12/12/11 - 12/16/11

Final ExamsWeek



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In class presentations will be graded according to the following rubric.

4 Points. Your solution is exemplary. Four points will only be awarded for solutionsto particularly difficult problems or very elegant solutions. Generally, 3 points will bethe most that you can earn for a solution.3 Points. Your solution is completely accurate and essentially perfect.2 Points. Your solutions is good, but incompletely developed. Correct solutions thatare not explained may received 2 points even if correct. You have the right idea, butthe solution is flawed.1 Points. Solution is sketchy. There is some correct work, but most of the solution isincorrect.0 Points. Solution is completely incorrect.


Fall 2011 Introduction to Modern Mathematics

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