§ 4. The Integers and the Real Numbers

Definition 4.1 : Binary Operation

A binary operation on a set \(A\) is a function \(f\) mapping \(A \times A\) into \(A\).

We usually write the value of the function at \((a, a')\) as \(afa'\).

Definition 4.2 : Real Numbers

We assume there exists a set \(\mathbb{R}\), called the set of real numbers, two binary operations \(+\) and \(\cdot\) on \(\mathbb{R}\), called the addition and multiplication operations, respectively, and an order relation \(<\) on \(\mathbb{R}\), such that the following properties hold:

1. \((x+y)+z=x+(y+z)\),
   \((x \cdot y) \cdot z=x \cdot(y \cdot z)\) for all \(x, y, z\) in \(\mathbb{R}\).

2. \(x+y=y+x\),
   \(x \cdot y=y \cdot x\) for all \(x, y\) in \(\mathbb{R}\).

3. There exists a unique element of \(\mathbb{R}\) called zero, denoted by \(0\), such that \(x+0=x\) for all \(x \in \mathbb{R}\).
   There exists a unique element of \(\mathbb{R}\) called one, different from \(0\) and denoted by \(1\), such that \(x \cdot 1=x\) for all \(x \in \mathbb{R}\).

4. For each \(x\) in \(\mathbb{R}\), there exists a unique \(y\) in \(\mathbb{R}\) such that \(x+y=0\).
   For each \(x\) in \(\mathbb{R}\) different from \(0\), there exists a unique \(y\) in \(\mathbb{R}\) such that \(x \cdot y=1\).

5. \(x \cdot(y+z)=(x \cdot y)+(x \cdot z)\) for all \(x, y, z \in \mathbb{R}\).

6. If \(x>y\), then \(x+z>y+z\).
   If \(x>y\) and \(z>0\), then \(x \cdot z>y \cdot z\).

7. The order relation \(<\) has the least upper bound property.

8. If \(x<y\), there exists an element \(z\) such that \(x<z\) and \(z<y\).

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From properties (1)-(5) follow the familiar "laws of algebra."
Given \(x\), one denotes by \(-x\) that number \(y\) such that \(x+y=0\); it is called the negative of \(x\). One defines the subtraction operation by the formula \(z-x=z+(-x)\).
Similarly, given \(x \neq 0\), one denotes by \(1 / x\) that number \(y\) such that \(x \cdot y=1\); it is called the reciprocal of \(x\). One defines the quotient \(z / x\) by the formula \(z / x=z \cdot(1 / x)\).
The usual laws of signs, and the rules for adding and multiplying fractions, follow as theorems. We often denote \(x \cdot y\) simply by \(x y\).

When one adjoins property (6) to properties (1)-(5), one can prove the usual "laws of inequalities".
We define a number \(x\) to be positive if \(x>0\), and to be negative if \(x<0\). We denote the positive reals by \(\mathbb{R}_{+}\)and the nonnegative reals by \(\overline{\mathbb{R}}_{+}\).
Properties (1)-(6) are familiar properties in modern algebra. Any set with two binary operations satisfying (1)-(5) is called by algebraists a field; if the field has an order relation satisfying (6), it is called an ordered field.

Properties (7) and (8), on the other hand, are familiar properties in topology. They involve only the order relation; any set with an order relation satisfying (7) and (8) is called by topologists a linear continuum.

Property (8) can be proved using properties (1)-(7), so only properties (1)-(7) are axioms.

Definition 4.3 : Inductive Set / Positive Integers

A subset \(A\) of the real numbers is said to be inductive if it contains the number \(1\), and if for every \(x\) in \(A\), the number \(x+1\) is also in \(A\). Let \(\mathcal{A}\) be the collection of all inductive subsets of \(\mathbb{R}\). Then the set \(\mathbb{Z}_{+}\) of positive integers is defined by the equation

\[ \mathbb{Z}_{+}=\bigcap_{A \in \mathcal{A}} A. \]

Theorem 4.0 : Property of Positive Integers

1. \(\mathbb{Z}_{+} \subset \mathbb{R}_{+}\).
2. \(1\) is the smallest element of \(\mathbb{Z}_{+}\).
3. \(\mathbb{Z}_{+}\) is inductive.
4. (Principle of induction) If \(A\) is an inductive set of positive integers, then \(A=\mathbb{Z}_{+}\).

Definition 4.4 : Integers / Rational Numbers

We define the set \(\mathbb{Z}\) of integers to be the set consisting of the positive integers \(\mathbb{Z}_{+}\), the number \(0\), and the negatives of the elements of \(\mathbb{Z}_{+}\).

The sum, difference, and product of two integers are integers, but the quotient is not necessarily an integer. The set \(\mathbb{Q}\) of quotients of integers is called the set of rational numbers.

Definition 4.5 : Section of Positive Integers

If \(n\) is a positive integer, we use the symbol \(S_{n}\) to denote the set of all positive integers less than \(n\); we call it a section of the positive integers. The set \(S_{1}\) is empty, and \(S_{n+1}\) denotes the set of positive integers between 1 and \(n\), inclusive. We also use the notation

\[ \{1, \ldots, n\}=S_{n+1} \]

for the latter set.

Theorem 4.1 : Well-Ordering Property

Every nonempty subset of \(\mathbb{Z}_{+}\)has a smallest element.

Proof

We first prove that, for each \(n \in \mathbb{Z}_{+}\), the following statement holds: Every nonempty subset of \(\{1, \ldots, n\}\) has a smallest element.

Let \(A\) be the set of all positive integers \(\mathrm{n}\) for which this statement holds. Then \(A\) contains \(1\), since if \(n=1\), the only nonempty subset of \(\{1, \ldots, n\}\) is the set \(\{1\}\) itself. Then, supposing \(A\) contains \(n\), we show that it contains \(n+1\). So let \(C\) be a nonempty subset of the set \(\{1, \ldots, n+1\}\). If \(C\) consists of the single element \(n+1\), then that element is the smallest element of \(C\). Otherwise, consider the set \(C \cap\{1, \ldots, n\}\), which is nonempty. Because \(n \in A\), this set has a smallest element, which will automatically be the smallest element of \(C\) also. Thus \(A\) is inductive, so we conclude that \(A=\mathbb{Z}_{+}\); hence the statement is true for all \(n \in \mathbb{Z}_{+}\).

Now we prove the theorem. Suppose that \(D\) is a nonempty subset of \(\mathbb{Z}_{+}\). Choose an element \(n\) of \(D\). Then the set \(A=D \cap\{1, \ldots, n\}\) is nonempty, so that \(A\) has a smallest element \(k\). The element \(k\) is automatically the smallest element of \(D\) as well.

Theorem 4.2 : Strong Induction Principle

Let \(A\) be a set of positive integers. Suppose that for each positive integer \(n\), the statement \(S_{n} \subset A\) implies the statement \(n \in A\). Then \(A=\mathbb{Z}_{+}\).

Proof

If \(A\) does not equal all of \(\mathbb{Z}_{+}\), let \(n\) be the smallest positive integer that is not in \(A\). Then every positive integer less than \(n\) is in \(A\), so that \(S_{n} \subset A\). Our hypothesis implies that \(n \in A\), contrary to assumption.

Theorem 4.3 : Archimedean Ordering Property

The set \(\mathbb{Z}_{+}\) of positive integers has no upper bound in \(\mathbb{R}\).

Proof

We assume that \(\mathbb{Z}_{+}\)has an upper bound and derive a contradiction. If \(\mathbb{Z}_{+}\)has an upper bound, it has a least upper bound \(b\). There exists \(n \in \mathbb{Z}_{+}\)such that \(n>b-1\); for otherwise, \(b-1\) would be an upper bound for \(\mathbb{Z}_{+}\) smaller than \(b\). Then \(n+1>b\), contrary to the fact that \(b\) is an upper bound for \(\mathbb{Z}_{+}\).