Concepts of Mathematics - Summer I 2012

Instructor Information

• Instructor: William Gunther
• Email: wgunther@math.cmu.edu
• Office: Wean 6211
• Office Hours:
• Monday 3:00pm - 4:00pm
• Tuesday 3:00pm - 4:00pm
• Wednesday 5:00pm - 6:00pm
• Thursday 11:30am - 12:30pm
• Friday 3:00pm - 4:00pm
• Sunday 2:00pm - 4:00pm
• By Appointment

Course Information

• Course Title: Concepts of Mathematics
• Course Number: 21-127
• Lecture Room: Wean 4623
• Lecture Time: 1:30-2:50pm every weekday
• Syllabus: PDF/LaTeX
• Text: There is no required text to purchase. The following are free things we will use.
• Optional Resources:
• Polya, George. How to Solve It. Optional; Available very cheaply on Amazon

Announcements

• Friday 5/25/12: The LaTeX tutorial will be Tuesday May 29 at 5-6:30ish pm in Wean 5207
• Sunday 5/20/12: Course calendar is settled (as far as I know!). Homework 1 is up as well, and notes for the first day. Notes for the second and third day should be done before class tomorrow.
• Thursday 5/17/12: Syllabus is finalized. I am still messing with the course calendar though. It will be settled as far as I know by the beginning of the course

Course Summary

Welcome to Concepts of Mathematics. This course is a prerequisite for any theoretical course here at CMU. It will train you how to reason and think about mathematical problems, as well as give you a base of knowledge and concepts that will aid in your understanding of more advanced mathematics. This course will have three parts, which should transition fairly well to one another.

The first part of the course will be on logic and proof techniques. Here we will introduce logical notation and talk about how to prove different statement. We will pay particular interest to quantifiers. We will also talk about the algebra of logic. The first part will conclude with some basics of set theory, which make our logic much more expressive.

The second part of the course will cover structures on sets. We talk about relations. Using relations, we will define orderings and equivalence relations. From equivalence relations, we will make a quick detour into the world of number theory. After this detour we will define maybe the most important type of relation: the function. We will discuss properties of functions, and cardinality, ending with Cantor's Theorem.

The third part of the course will be on discete math. Ostensibly, this is a complete diversion, but when we begin with counting; here we borrow from the notions of injections and surjections from the previous part to ensure that we neither over or under count. After learning to count we look at graph theory, which relates to our study of combinatorics and our prior study of relations.

This course, especially over the summer when the time to teach is cut in more than half, is very intense. As you can see, I have scheduled many office hours. The homeworks are designed to be very challenging, yet incrementally more challenging to allow you to ease into a topic and do the homework as we learn.

Course Calendar

Monday Tuesday Wednesday Thursday Friday
21
Notes:
Puzzles and Numbers
22
Notes:
Inequalities
23
Notes:
Basic Logic and Proofs
24
Notes:
Algebra of proofs
25
Notes:
HW1:
Quantifiers and proofs
28
Memorial Day
No Class
29
Notes:
HW2:
Proof Techniques and Induction
30
Notes:
Natural Number Induction
31
Notes:
Induction/Basic Set Theory
1
Notes:
HW3:
Comprehension
4
Exam 1
Tech Q's:
5
Notes:
More Set Theory
6
Notes:
Relations
7
Notes:
Equivalence Relations
8
Notes:
HW4:
Modular Arithmetic
11
Notes:
Basic Functions
12
Notes:
HW5:
Properties of Functions
13
Notes:
Cardinality
14
Notes:
Infinite Sets
15
Notes: PDF
HW6:
Cantor's Theorem
18
Exam 2
Tech Q's:
19
Notes:
Basic Counting
20
Notes:
21
Notes:
Counting Proofs
22
Notes:
HW7:
Pigeon-Hole Principle
25
Notes:
Inclusion Exclusion
26
Notes: PDF
Basic Graph Theory and Trees
27
Notes: PDF
HW8:
Coloring
28
Notes: PDF
Ramsey Theory
29
Exam 3
Tech Q's:

Note: to use any of the LaTeX, you will need the preamble I import: preamble.tex