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Appendix A Definitions for this Workbook

These definitions are typically introduced in introductory courses on proofs, discrete math, and number theory. They are useful for completing the proof exercises in this workbook and valuable for any aspiring mathematician to remember. However, memorizing these definitions is not as important for your success in real analysis as is knowing how to use definitions in proofs (both direct and indirect proofs). So, you may want to focus more on applying these definitions than on memorizing them!

Definition A.14. Rational Numbers.

The set of rational numbers consists of the ratios of integers, whose denominator is nonzero:

\begin{equation*} \mathbb{Q} = \bigl\{ \frac{p}{q} \quad \colon \quad p,q\in \mathbb{Z}, q\neq 0 \bigr\} . \end{equation*}

Note that Definition A.14 defines the rational numbers as a set. Their algebra would require us to specify that two fractions represent the same rational number exactly when their cross-ratios are equal (they have the same “lowest terms”):

\begin{equation*} \mathbb{Q} = \bigl\{ \frac{p}{q} \; \colon \; p,q\in \mathbb{Z}, q\neq 0 \bigr\} / \sim \text{ where } \frac{p}{q} \sim \frac{p^\prime}{q^\prime} \text{ whenever } pq^\prime = p^\prime q. \end{equation*}
Definition A.15. Parity, Even, Odd.

Let \(n\in \mathbb{Z}\) be an integer. We say that the parity of \(n\) is either even or odd depending on whether:

\begin{align} \text{There exists }k\in\mathbb{Z}\text{ such that } && n &= 2k & \text{(}n\text{ is even)}\label{def_even}\tag{A.3}\\ \text{There exists }k\in\mathbb{Z}\text{ such that } && n &= 2k+1 & \text{(}n\text{ is odd)}\label{def_odd}\tag{A.4} \end{align}

Mostly, Definition A.15 is helpful for comparing two integers. For example, to say that “\(x\) and \(y\) have the same parity” is a short way to say “\(x\) and \(y\) are either both even or both odd”.

Definition A.16. Congruence Modulo \(p\).

Let \(m,n\in\mathbb{Z}\) be two integers, and \(p\in\mathbb{N}\) be a natural number. We say that \(m\) and \(n\) are congruent modulo \(p\) (or “mod \(p\)” for short) if their difference is a multiple of \(p\text{:}\)

\begin{equation*} \text{There exists }k\in\mathbb{Z}\text{ such that } m-n = kp. \end{equation*}

This generalizes the notion of parity: in particular, two integers have the same parity if and only if they are congruent modulo 2. (Can you prove this assertion?)

Definition A.17. Divisibility.

Let \(n,p\in\mathbb{Z}\) be two integers. We say that \(n\) divides \(p\) (or, equivalently, that \(p\) is a multiple of \(n\)) if:

\begin{equation*} \text{There exists }k\in\mathbb{Z}\text{ such that } kn = p. \end{equation*}

When this is true, we use the notation \(n \mid p\text{.}\)

The preceding two definitions are also related: two integers \(m,n\) are congruent modulo \(p\) if, and only if, \(p \mid (m-n)\text{.}\)

Definition A.18. Factorial.

For all natural numbers \(n\) we define \(n\) factorial to be

\begin{equation*} n! = n \cdot (n-1) \cdot (n-2) \cdot \ldots \cdot 1, \end{equation*}

by multiplying all natural numbers from 1 up to \(n\text{.}\)