Logic and Mathematical Reasoning

Lecture notes

Syllabus

Course description

Ever since the reformulation of calculus in the middle of the 19^th^ century, mathematicians have been insistent upon maintaining a high standard of rigor. This led to the development of formal logic in the early 20^th^ century and using set theory as a foundation of mathematics around the same time. In this course you will learn the basics of classical logic including propositions, connectives, implications and quantifiers, as well as basic set theory including relations, functions and cardinality. You will also get a brief introduction to combinatorics and graph theory. In addition to the context provided by this material, you will be learning how to read, analyze and write proofs. Believe it or not, proofs are the bread and butter of mathematics whereas almost all your prior coursework has focused on computation. Computation is an important part of mathematics, especially applied mathematics, statistics and cryptography, but it is only a piece the entirety of mathematical thought. Learning to think logically, read and write proofs requires a rewiring of the way you think, and this is potentially a challenging course for that reason. However, you should commit to devoting serious time to learning the material and how to think like a mathematician because it will pay off in the long run, both in your mathematical studies, and in your future career.

We will use A Transition To Advanced Mathematics, 7^th^ ed., by Douglas Smith, Maurice Eggen, and Richard St.~Andre.

Course objectives

At the conclusion of this course, students should be able to:

• translate between semi-formal mathematics and logical formulas;
• construct proofs using the following techniques: direct, contrapositive, contradiction, induction, and combinatorial;
• construct proofs involving basic number theory (divisibility, gcd, parity, rational numbers), sets, relations and partitions, functions (including injective and surjective properties), cardinality, and basic graph theory;
• evaluate correctness of proofs and extract the key idea;
• perform computations involving logic, sets, functions, and graphs.

Course Outline And Topics

1. Logic and Proof $\approx$ 6 classes

   1. Propositions and Connectives
   2. Conditionals and Biconditionals
   3. Quantifiers
   4. Basic Proof Methods I
   5. Basic Proof Methods II
   6. Proofs Involving Quantifiers
   7. Additional Examples of Proofs

2. Set Theory $\approx$ 9 classes

   1. Basic Concepts of Set Theory
   2. Set Operations
   3. Extended Set Operations and Index Families of Sets
   4. Induction
   5. Principles of Counting

3. Relations $\approx$ 6 classes

   1. Cartesian Products and Relations
   2. Equivalence Relations
   3. Partitions
   4. Graphs

4. Functions $\approx$ 4 classes

   1. Functions as Relations
   2. Constructions of Functions
   3. Functions That Are Onto; One-to-One Functions
   4. Images of Sets

5. Cardinality $\approx$ 2 classes

   1. Equivalent Sets; Finite Sets
   2. Infinite Sets

Expectations

I am maintaining notes on my website, a link to which you can find both at the top of this syllabus and on Blackboard. You can access them as html, but a pdf is also available. These notes are not substitutes for attending class.

You are expected to read ahead in the book before each class so that you are primed to learn about the material in more depth. I used the word primed intentionally: you should think of this reading ahead as being painted with primer so that when you get to class the lecture (i.e., the paint) sticks effectively. The purpose of reading ahead is not to learn everything on your own.

Grading procedure

• Two midterm exams (20% each)
• Homework assignments (25% total)
• Attendance and participation (5% total)
• Comprehensive final exam (30%)

Important dates

Makeups

If an emergency arises which requires you to miss an exam, I must be made aware at least two hours prior to the start time of your exam. Note: proper documentation will be required before a makeup arrangement is considered.

Academic dishonesty/misconduct

Academic misconduct includes, but is not limited to, cheating, plagiarism and forgery, and soliciting, aiding, abetting, concealing, or attempting such acts. Plagiarism may consist of copying, paraphrasing, or otherwise using written or oral work of another without proper acknowledgment of the source or presenting oral or written material prepared by another as ones own. At minimum, cheating will result in that assignment receiving a grade of zero.

Official communication

Your SIUE student e-mail account is the official method to communicate between you and your instructor. Official communication will not be sent to your personal e-mail (yahoo, wildblue, gmail etc.).

Disability Support Services

If you have a documented disability that requires academic accommodations, please go to Disability Support Services for coordination of your academic accommodations. DSS is located in the Student Success Center, Room 1270; you may contact them to make an appointment by calling (618) 650-3726 or sending an email to disabilitysupport@siue.edu. Additional information is located online at http://siue.edu/dss.

Disclaimer

This syllabus is subject to change by the instructor if deemed necessary for the benefit of student learning or to correct errors and omissions.