# 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.