Concepts of Mathematics - Summer II 2012

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Note: My friend Will Gunther designed this pretty-looking site. He is a genius.

Instructor Information

  • Instructor: Brendan Sullivan
  • Email: bwsulliv@andrew.cmu.edu
  • Office: Wean 7104
  • Office Hours (also by appointment via email):
    • MTWRF: right after class until 11:00ish
    • Monday 2-4pm
    • Thursday 1-4pm
    • Saturday 2-5pm
  • T.A.: Lisa Espig
  • Email: lespig@andrew.cmu.edu
  • Office: Wean 7207
  • Office Hours:
    • Monday 4-6pm
    • Wednesday 3-6pm
    • Thursday 4-7pm
    • Sunday 3-6pm

Course Information

  • Course Title: Concepts of Mathematics
  • Course Number: 21-127
  • Lecture Room: Scaife 125
  • Lecture Time: 9:00am-10:20am MTWRF
  • Syllabus: PDF / Web
  • Texts: There is no required text to purchase. I will provide condensed lecture notes (downloadable from the course calendar below) and textbook materials written by me! (viewable in a .pdf app, more details in that link)
  • Optional Resources:
    • Day, Martin. An Introduction to Proofs and the Mathematical Vernacular (free .pdf)
    • Polya, George. How to Solve It. (Available very cheaply on Amazon)
    • Solow, Daniel. How to Read and Do Proofs. (Cheap used copies on Amazon)
    • D'Angelo & West. Mathematical Thinking: Problem-solving and Proofs.
      (Previous standard text for this course, more expensive on Amazon)
    • Velleman, Daniel. How To Prove It: A Structured Approach.
      (Cheap paperback copies on Amazon)


Groups

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Announcements

Course Summary

Welcome to Concepts of Mathematics! This course is a prerequisite for any theoretical course here at CMU in mathematics, computer science, and the natural sciences. It will train you how to reason and think about mathematical problems, as well as give you a base of knowledge, techniques, and mathematical fluency that will aid in your understanding of more advanced mathematics.

See the course syllabus for more information about Concepts of Mathematics, in general, and a description of this summer session (content, assessment, policies, etc.), in particular.

Useful Links

Course Calendar

Note: to use any of the LaTeX files I provide, you will need the preamble I import: preamble.tex
In the preamble of your .tex file, include the line "\input{/*****/preamble.tex}", where ***** is the directory path to where preamble.tex is stored on your computer.


NOTE: None of these links work anymore because the course is over. Sorry! I gotta keep my secrets, ya know?

Monday Tuesday Wednesday Thursday Friday
2
Notes: PDF
Intro + Syllabus
What is math?
Puzzles
3
Notes: PDF
What is a proof?
Axioms
4
Reqd. Reading: Chapter 2

NO CLASS!
Independence Day
5
Notes: PDF
Inductive arguments
Sets
Examples
6
Notes: PDF
HW1 due! PDF/LaTeX
(Picture needed to compile
Set Operations
Power sets
9
Notes: PDF
Set equalities
Logical Statements
10
Notes: PDF
HW2 due! PDF/LaTeX
Connectives
Conditionals
Logical Negations
11
Notes: PDF
Proof Strategies
Proof Examples
12
Notes: PDF
Using proof strategies
Workshop
13
Notes: PDF
HW3 due! PDF/LaTeX
Induction
Workshop
16
Exam 1!

Prep Q's: PDF
17
Notes: PDF
More on Induction
Generalized strategies
Structural Induction
18
Notes: PDF
Relations
Examples
Properties
19
Notes: PDF
Equivalence Relations
Equivalence Classes
Modular Arithmetic
20
Notes: PDF
HW4 due! PDF/LaTeX
E1 Extra Credit due: PDF/LaTeX
Number Theory
Primes
23
Notes: PDF
Functions
Examples
Images + Preimages
24
Notes: PDF
HW5 due! PDF/LaTeX
Injections + Surjections
Proof Strategies
Bijections
25
Notes: PDF
Composing Functions
Inverses
26
Notes: PDF
Cardinality
Finite Sets
Countable Sets
27
Notes: PDF
HW6 due! PDF/LaTeX
Infinite Sets
Uncountable Sets
Cantor's Argument
30
Exam 2!

Prep Q's: PDF/LaTeX
31
Notes: PDF
Counting
Rule of Sum
Rule of Product
1
Notes: PDF
Counting Objects
Proof Strategies
Combinatorial Identities
2
Notes: PDF
Counting in 2 ways
Reading Proofs
Pirates & Gold
3
Notes: PDF
HW7 due! PDF/LaTeX
Pigeonhole Principle
Inclusion/Exclusion
6
Notes: PDF
Probability Theory
Spaces and Functions
Results
7
Notes: PDF
Conditional Probablity
8
Notes: PDF
HW8 due! PDF/LaTeX
Graph Theory
Definitions
Properties
9
Notes: PDF
Graph Colorings
Probabilistic Method
Ramsey Theory
10
Exam 3!

Prep Q's: PDF/LaTeX