Prerequisite: Math 76 (Calculus II).

Hours: 4.

Catalog description: Introduction to the language and problems of mathematics. Use of LaTeX as a typesetting tool. Topics include set theory, symbolic logic, types of proofs, and mathematical induction. Emphasis on students constructing, explaining, and justifying mathematical arguments through active learning.

Course goals
Upon completion of this course, students should know/understand:

• Importance of a formal, rigorous proof
• When it is sufficient to give an example/counterexample and when an arbitrary element of a set must be considered
• Set operations (intersection, union, difference, complement) and their properties
• Meaning of logical operations (and, or, negation, implication) and quantifiers (universality and existence)
• Difference between different types of proof (direct, by contrapositive, by contradiction, by example/counterexample, etc.)
• Divisibility properties of integer numbers; congruences; basic properties of rational and irrational numbers
• The notions of a relation, equivalence relation; properties of equivalence classes
• The notions and properties of one-to-one, onto, bijective, inverse functions
• Principle of Mathematical Induction

Learning outcomes
Upon completion of this course, students will be able to do:

• Recognize simple statements that can be proved using one of the following techniques, and prove them:
  • direct proof
  • proof by contrapositive
  • proof by contradiction
  • vacuous proof
  • trivial proof
  • proof by cases
  • proof by example/counterexample
  • proob by Induction
• Determine the truth values of simple statements.
• Type mathematical texts in LaTeX.

Grading policy: Your grade for the course will be based on your performance on exams and homework (more information on these below). The number of points awarded for these is as follows.

Class attendance is strongly encouraged and expected. In addition to new material, important course information such as homework assignments will be given, homework may be discussed, and sometimes handouts may be distributed in class. In addition to attending class, you should expect to study at least 8 hours a week.

There will be weekly homework due every Wednesday at 11:59 pm. Be prepared to spend a few hours on each homework (you will learn to write mathematical proofs, polish your proofs, and type text in LaTeX - all of this takes time). You should write complete solutions, with all necessary proofs, and specific examples and counterexamples when appropriate, for each problem. Unsupported answers may receive zero credit. All proofs must be rigorous (use the style of proofs given in class and in the book). You should do homework in groups of 2-4 people. The groups do not have to be the same for all homeworks: you may do one homework with one person/group and another homework with another. However, in case you want to change your group, you should notify your group members well in advance (at least two weeks notice is recommended.) If you do not have a group, please let me know, and I will help you find one. Write the names of all group members on the first page clearly. Please make sure that everyone in the group understands and agrees with all solutions. The primary goal of homework is to learn, so if someone else does a problem, you don't learn anything. You need to actively participate to make sure you learn well. The whole group will receive the same grade. If you have any questions, or need someone to listen to and possibly comment on your ideas, please do not hesitate to ask your instructor (this is what her office hours are for!) Also, you are encouraged to work with any of your classmates even if they are not in your group. However, your group has to write all solutions by yourself, in your own words. Any copying will be considered cheating and will not be tolerated. Homework has to be typed in LaTeX and submitted through overleaf. Please share your homework file before the deadline (11:59 pm), and give your instructor "can edit" permissions. No late submissions will be accepted unless you are sick or have another serious reason to delay homework (notify your instructor in advance, or provide doctor's note, etc.) Tip: share your overleaf file with your instructor as soon as you start working on your homework. It will not be checked or graded before the deadline, but that way you will be sure that you turned it in on time. Continue working on your homework at any time before the deadline, however, you may not make any changes after the deadline. If any edits are made after the deadline, this will be considered cheating, and your homework will not be accepted.

Exams: There will be three tests (50 minutes long each) and one final comprehensive exam (2 hours long). If for any reason you can not take a test at the scheduled time, please let me know as soon as possible, and certainly before the test. See schedule for exam dates.

Extra Help: It is essential not to fall behind, because most classes will use the material studied previously. If you have trouble with some material, seek help in the following ways:

• Ask me, either in class or privately. Don't be shy to ask questions. If you don't understand something, chances are very high that somebody else doesn't understand that either. So your classmates will be thankful to you for asking questions in class! My contact info is given on the course web page.
• Attend office hours. These are drop-in hours when I am in my office for sure, with my door open. If my posted office hours do not work for your schedule, make appointments. Again, my contact info is on the course web page. Questions about homework are welcome at any time. However, after you got help from me, you should put aside all the notes you made while talking to me, and write a complete solution from scratch, in your own words. I rarely write complete sentences during my office hours, I may only help you to find an approach, an idea. You are responsible for writing a complete solution. Just copying formulas and pictures from my white board and notes will receive zero credit.
• Work with your classmates. Note: working on your homework together is encouraged, however, every group should write down their solutions on their own. If somebody is helping you with your homework then the same rules as for my helping you apply, i.e. you should put aside all the notes and write a complete solution from scratch, in your own words. That way you will learn the best.
• Do also problems that were not assigned. Use the answers and hints in the back of the book to check your solutions.
• Attend the tutor lab. The Math Department tutor lab schedule is posted here.

Students with disabilities: upon identifying themselves to the University, students with disabilities will receive necessary accommodation for learning and evaluation. For more information, see http://www.fresnostate.edu/studentaffairs/ssd/

Academic honesty: cheating and plagiarism will not be tolerated in this class. For information on the University's policy, see the University Catalog (section Policies and Regulations).

Honor Code: Members of the CSU Fresno academic community adhere to principles of academic integrity and mutual respect while engaged in university work and related activities. You should:

• understand or seek clarification about expectations for academic integrity in this course (including no cheating, plagiarism and inappropriate collaboration)
• neither give nor receive unauthorized aid on examinations or other course work that is used by the instructor as the basis of grading.
• take responsibility to monitor academic dishonesty in any form and to report it to the instructor or other appropriate official for action.

Computers: At California State University, Fresno, computers and communications links to remote resources are recognized as being integral to the education and research experience. Every student is required to have his/her own computer or have other personal access to a workstation (including a modem and a printer) with all the recommended software. The minimum and recommended standards for the workstations and software, which may vary by academic major, are updated periodically and are available from Information Technology Services or the University Bookstore. In the curriculum and class assignments, students are presumed to have 24-hour access to a computer workstation and the necessary communication links to the University's information resources.

Disruptive Classroom Behavior: student conduct which disrupts the learning process will not be tolerated and may lead to disciplinary action and/or removal from class.

Copyright policy: Copyright laws and fair use policies protect the rights of those who have produced the material. The copy in this course has been provided for private study, scholarship, or research. Other uses may require permission from the copyright holder. The user of this work is responsible for adhering to copyright law of the U.S. (Title 17, U.S. Code). Digital Campus course web sites contains material protected by copyrights held by the instructor, other individuals or institutions. Such material is used for educational purposes in accord with copyright law and/or with permission given by the owners of the original material. You may download one copy of the materials on any single computer for non-commercial, personal, or educational purposes only, provided that you (1) do not modify it, (2) use it only for the duration of this course, and (3) include both this notice and any copyright notice originally included with the material. Beyond this use, no material from the course web site may be copied, reproduced, re-published, uploaded, posted, transmitted, or distributed in any way without the permission of the original copyright holder. The instructor assumes no responsibility for individuals who improperly use copyrighted material placed on the web site.

Subject to Change: This syllabus and schedule are subject to change in the event of extenuating circumstances. If you are absent from class, it is your responsibility to check on announcements made while you were absent.