Prerequisite: Math 76 (Calculus II).

Hours: 3.

Content: Mathematical notations, proofs (what they are, types of proofs, techniques), introduction to logic, equivalence relations, basic facts from set theory, number theory, and calculus, and their proofs.

See also the Catalog Description.

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.

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.

Ammendment (made after Test 2): if you get an A on both Test 3 and Final Exam, you get an A in the course; if you get a B or an A on both Test 3 and Final Exam, but not both As, you get a B in the course.

Warning: While I do want to encourage you to study for the exams, in practice, very few people ever take advantage of the above ammendment. That is, those who, for example, get an A on both the last test and Final exam, usually would have gotten an A in the course anyway, even without the above rule. So don't count on this ammendment too much and keep working on your homework and try to get you total score (as well as your knowledge) as high as possible.

Exams: There will be three tests (1 hour 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. Solutions to past exams will be posted there in the future.

Class attendance is required during the first two weeks and strongly encouraged after that. In addition to new material, important course information such as homework assignments will be given, homework will be collected, and sometimes handouts will be distributed in class.

There will be weekly homework due Thursday at the beginning of the class period (9:30 am). Up to 3 times during the semester, you may submit your homework one class day late (i.e. on Tuesday of the next week, at 9:30 am, but not later than then because I will try to return graded papers and often hand out solutions at our next class meeting). However, if you are sick or have an emergency, and unable to come to campus, please notify me and ask for a longer extension.
You should write a complete solution, 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).

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. My contact info is given on the course web page. 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 allowed, however, every student should write down their solutions on their own. Also, if you have worked with someone, please indicate that on your paper. 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.
• Do also problems that were not assigned. Use the answers and hints in the back of the book to check your answers.

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://studentaffairs.csufresno.edu/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).

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.