Preliminaries and review

In order to make progress in Math 3230, it is crucial to be up-to-date with the material covered in Math 2710.

The Book of Proof is a free textbook (used in many 2710 sections) that will be valuable reference for the foundations of this course. In particular, you should review the following topics.

Sets

We will assume that you are familiar with the basic language of sets, including

• Set-builder notation
• The basic operations of union, intersection, difference
• Cartesian products

These ideas are covered in Chapter 1 of Book of Proof and are covered in these video lectures.

Logic

We will rely on the basic principles of logic as outlined in Chapter 2 of Book of Proof. In particular, make sure that you are comfortable with the universal (“for all”) and existential (“there exists”) quantifiers and how to interpret statements of the form:

• “For all x in A, there exists y in B”
• “There exists x in A, such that for all y in B”

Also make sure you know how to negate this type of statement.

A set of video lectures on these topics are available here. Quantifiers are discussed starting here and negation is discussed starting here.

Mathematical Induction

We will make frequent use of arguments by mathematical induction. This is discussed in Chapter 10 of Book of Proof and video lectures on this topic are available here

Congruence

One of our first and most important examples of groups will require use of the idea of “congruent integers.” This is discussed in Chapter 5, Section 2 of Book of Proof. Video discussion is available here and here with additional information in Chapter 11.2 about 21 minutes in.

Functions, relations, inverse functions

We will make use of the idea of “function” and “relation.” This is discussed at length in Chapters 11 and 12 of Book of Proof. A set of video lectures covering these chapters is available here:

Of particular importance are:

• the idea of an equivalence relation, in Chapter 11.3 of Book of Proof with a video Content here. The related idea of a partition of a set is discussed in Chapter 11.4 with video here.
• the idea of inverse functions, in Chapter 12.5, with video here

Euclid’s Algorithm

Euclid’s algorithm will be an important tool in working with certain examples. See Book of Proof Chapter 7.3 and the numerical example.

The Binomial Theorem

The binomial theorem is a standard tool and is discussed in Book of Proof Chapter 3.5 with video here.