Problem 27 - A History of Mathematics: An Introduction

Question

Define an addition on Dedekind's cuts. Show that \(\alpha+\beta=\) \(\beta+\alpha\) for any two cuts \(\alpha\) and \(\beta\).

Step-by-Step Solution

Answer

Based on the given step-by-step solution, define an addition operation on Dedekind's cuts and prove that this operation is commutative. The addition operation on Dedekind's cuts, \(\alpha\) and \(\beta\), can be defined as the cut \(\alpha + \beta = (\) \(A_{\alpha+\beta}\), \(B_{\alpha+\beta}\)) such that \(A_{\alpha+\beta} = \{ a+b | a \in A_\alpha \text{ and } b \in A_\beta \}\) and \(B_{\alpha+\beta} = \mathbb{Q} \setminus A_{\alpha+\beta}\). To prove the commutativity of the addition operation on Dedekind's cuts, we need to show that for any two cuts \(\alpha\) and \(\beta\), \(\alpha+\beta = \beta+\alpha\). This can be achieved by showing that \(A_{\alpha+\beta} = A_{\beta+\alpha}\) and consequently, \(B_{\alpha+\beta} = B_{\beta+\alpha}\). We define this operation by taking every possible sum of an element from \(A_\alpha\) and an element from \(A_\beta\) to create \(A_{\alpha+\beta}\), and the rest of the possible sums to create \(B_{\alpha+\beta}\). Since the addition of rational numbers is commutative, we can conclude that the addition operation on Dedekind's cuts is also commutative, as both \(A_{\alpha+\beta}\) and \(B_{\alpha+\beta}\) are equal for any two cuts \(\alpha\) and \(\beta\).

Step 1: Understanding Dedekind's Cuts

Dedekind's cut is an approach to construct the real numbers using rational numbers. A Dedekind cut is a partition of the set of rational numbers into two non-empty sets \(A\) and \(B\) such that: 1. Every element in \(A\) is less than every element in \(B\). 2. \(A\) has no greatest element. In this context, a real number corresponds to a unique Dedekind cut. For example, the square root of 2 can be represented as a cut where \(A\) is the set of all rational numbers with a square smaller than 2, and \(B\) is the set of all rational numbers with a square greater than or equal to 2.

Step 2: Defining the Addition Operation on Dedekind's Cuts

Given two Dedekind's cuts \(\alpha\) and \(\beta\), let's represent their corresponding partitions as (\(A_\alpha\), \(B_\alpha\)) and (\(A_\beta\), \(B_\beta\)) respectively. We can define the sum of these two cuts, \(\alpha + \beta\), as the cut (\(A_{\alpha+\beta}\), \(B_{\alpha+\beta}\)) where: \(A_{\alpha+\beta} = \{ a+b | a \in A_\alpha \text{ and } b \in A_\beta \}\) \(B_{\alpha+\beta} = \{ a+b | a \in A_\alpha \text{ and } b \in B_\beta \} \cup \{ a+b | a \in B_\alpha \text{ and } b \in A_\beta \} \cup \{ a+b | a \in B_\alpha \text{ and } b \in B_\beta \}\) Essentially, we are taking every possible sum of an element from \(A_\alpha\) and an element from \(A_\beta\) to create \(A_{\alpha+\beta}\), and the rest of the possible sums to create \(B_{\alpha+\beta}\).

Step 3: Proving the Commutativity of the Addition Operation

To show that the addition operation defined on Dedekind's cuts is commutative, we need to prove that for any two cuts \(\alpha\) and \(\beta\), \(\alpha+\beta = \beta+\alpha\). This is equivalent to showing that \(A_{\alpha+\beta} = A_{\beta+\alpha}\) and \(B_{\alpha+\beta} = B_{\beta+\alpha}\) (since cuts are uniquely determined by their \(A\) sets). Consider an arbitrary element \(x \in A_{\alpha+\beta}\). By definition, \(x = a + b\) for some \(a \in A_\alpha\) and \(b \in A_\beta\). Since addition of rational numbers is commutative, \(x = b + a\) for the same values of \(a\) and \(b\). But this means that \(x \in A_{\beta+\alpha}\). Therefore, \(A_{\alpha+\beta} \subseteq A_{\beta+\alpha}\). Conversely, let \(x \in A_{\beta+\alpha}\). Then \(x = b + a\) for some \(a \in A_\alpha\) and \(b \in A_\beta\). Since addition of rational numbers is commutative, \(x = a + b\) for the same values of \(a\) and \(b\). This implies that \(x \in A_{\alpha+\beta}\). Hence, \(A_{\beta+\alpha} \subseteq A_{\alpha+\beta}\). Since \(A_{\alpha+\beta} \subseteq A_{\beta+\alpha}\) and \(A_{\beta+\alpha} \subseteq A_{\alpha+\beta}\), we have \(A_{\alpha+\beta} = A_{\beta+\alpha}\). Then, for the \(B\) sets: Since every rational number is either in the \(A\) set or the \(B\) set of a cut, we have \(B_{\alpha+\beta} = \mathbb{Q} \setminus A_{\alpha+\beta}\) and \(B_{\beta+\alpha} = \mathbb{Q} \setminus A_{\beta+\alpha}\). Since we've just shown that \(A_{\alpha+\beta} = A_{\beta+\alpha}\), it follows that \(B_{\alpha+\beta} = B_{\beta+\alpha}\). Finally, because both \(A\) and \(B\) sets are equal, we have \(\alpha + \beta = \beta + \alpha\), which proves the commutativity of the addition operation on Dedekind's cuts.

Key Concepts

Commutative Property

In mathematics, the commutative property is an essential concept that pertains to certain operations like addition. It tells us that the order in which we perform certain operations does not affect the final result. Specifically for addition, this property ensures that when two numbers are added together, the sum is the same regardless of their order:

• For any two numbers, say \(a\) and \(b\), we have \(a + b = b + a\).

This property is advantageous because it simplifies computations and makes manipulating expressions much easier. When working with complex structures like Dedekind's cuts, the commutative property can help verify that operations behave consistently—such as showing that the addition of cuts yields the same result regardless of the order.

Constructing Real Numbers

The journey to understanding real numbers involves discovering that they can be constructed from rational numbers. One effective method for this is through Dedekind's cuts. Dedekind's cuts partition the set of rational numbers into two non-empty sets:

• The first set contains numbers less than the real number being represented, often denoted as \(A\).
• The second set contains numbers greater than or equal to the real number, denoted as \(B\).

These partitions help in filling in the "gaps" between rational numbers, effectively creating real numbers. For instance, while you can't pinpoint a rational number for the square root of 2 exactly, you can use Dedekind's cuts to express this number by constructing an appropriate set of all rationals whose squares are less than 2.

Rational Numbers

Rational numbers are the building blocks of Dedekind's cuts and are defined as numbers that can be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers, and \(q eq 0\). They form an integral part of our number system due to their dense nature—between any two rational numbers, you can find another rational number. This density property is crucial when constructing real numbers via Dedekind's cuts. Although rational numbers are numerous, they do not cover all real numbers, leading to the need for Dedekind's formulation to accommodate irrationals. Thus, rational numbers serve as an essential framework around which the real number system is built.

Addition Operation

The addition operation in mathematics is a fundamental operation that combines two or more quantities. When dealing with Dedekind's cuts, addition has its own rules. Given two Dedekind cuts, \(\alpha\) and \(\beta\), you define their addition as constructing a new cut:

• The \(A\) part of this new cut, \(A_{\alpha + \beta}\), consists of all possible sums of one element from \(A_\alpha\) and one from \(A_\beta\).
• The \(B\) part is formed by considering all other combinations that do not belong to \(A_{\alpha+\beta}\).

This methodically constructed operation ensures the representation maintains coherent properties analogous to regular addition especially regarding the commutativity, hence \(\alpha + \beta = \beta + \alpha\). This structured approach allows us to manage real numbers' arithmetic using Dedekind's frameworks.