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(Calkin-Wilf tree) The binary tree in the plane has a distinguished vertex as 'root', and is constructed inductively. The root is joined to two new vertices; and each new vertex is then joined to two further new vertices \(-\) with the construction process continuing for ever (Figure 11 ). Label the vertices of the binary tree with positive fractions as follows: \- the root is given the label \(\frac{1}{1}\) \- whenever we know the label \(\frac{i}{j}\) of a 'parent' vertex, we label its 'left descendant' as \(\frac{i}{i+j},\) and its 'right descendant' \(\frac{i+j}{j}\). (a) Prove that every positive rational \(\frac{r}{s}\) occurs once and only once as a label, and that it occurs in its lowest terms. (b) Prove that the labels are left-right symmetric in the sense that labels in corresponding left and right positions are reciprocals of each other. \(\triangle\)

Step by step solution

01

Understand the construction

The tree begins with a single vertex labeled \(\frac{1}{1}\). Each vertex is expanded to two children with labels based on the parent's label \(\frac{i}{j}\). The left child receives the label \(\frac{i}{i+j}\), and the right child gets \(\frac{i+j}{j}\). This creates a unique labeling scheme as you move to new generations.

02

Show that every positive rational number appears

Assume \(\frac{r}{s}\) is any positive rational number in its reduced form. Through induction and properties of the Calkin-Wilf construction, every rational number generated is reduced because any common factors in \(i\) or \(j\) would exist between their sum \(i+j\) and one of the previous numbers. Therefore, each label \(\frac{r}{s}\) appears once as you move along paths determined uniquely by binary choices (left or right) from a parent to its children.

03

Prove the uniqueness of representation

Since the function to create children (\(\frac{i}{i+j}\) and \(\frac{i+j}{j}\)) only produces numbers with \(gcd(i+j, i) = 1\) and \(gcd(i+j, j) = 1\), each fraction will be presented in lowest terms. Furthermore, two paths cannot lead to the same label because the binary directional decision at each vertex (left or right) ensures each fraction \(\frac{r}{s}\) has a unique path derived from \(\frac{1}{1}\).

04

Demonstrate left-right symmetry based on recursion

Observe that for a parent \(\frac{i}{j}\), the left child \(\frac{i}{i+j}\) and right child \(\frac{i+j}{j}\), are such that the product, \(\frac{i}{i+j} \times \frac{i+j}{j} = \frac{i}{j}\). Thus, reciprocals are naturally formed across these symmetric positions in subtrees. This can be extended recursively, demonstrating a globally symmetric property across the tree.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Binary Trees

At the core of a Calkin-Wilf tree is the concept of a binary tree. A binary tree is a simple data structure used in computer science and mathematics where each node connects to two children, often referred to as the "left child" and "right child." This structure grows in a branching pattern as each node splits into two. This corresponds naturally to a tree's generational growth pattern.

• Every node in a binary tree can have up to two children.
• The growth continues indefinitely, given the nodes always have two children.
• In the context of the Calkin-Wilf tree, the nodes are labeled with fractions, starting from the root labeled as \(\frac{1}{1}\).

Understanding how binary trees are structured is key to appreciating how the Calkin-Wilf tree assigns unique fractions to each node.

Exploring Positive Fractions

Positive fractions are fractions where both the numerator and the denominator are positive integers. They represent parts of a whole in mathematics and are positioned between 0 and 1 on a number line. The fascinating part of the Calkin-Wilf tree is how it translates these fractions into a structured form within the tree.

• The labeling starts with the root, \(\frac{1}{1}\), and continues through left and right children.
• Every left child is created by moving fractions downward in value, and right children move them upward.
• The construction of the tree ensures each fraction is represented in its simplest form. This is critical as it guarantees uniqueness without redundancy.

Mastering positive fractions helps in comprehending the orderly display and sequence of values within the Calkin-Wilf tree.

Understanding Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, with a non-zero denominator. In simplest terms, any integer or fraction can be considered a rational number. The Calkin-Wilf tree presents an intriguing method of listing each positive rational number exactly once in its simplest form, which is a remarkable feat.

• Each label \(\frac{r}{s}\) on the tree represents a unique rational number.
• The construction of the Calkin-Wilf tree uses a binary decision at each node to systematically cover all positive rational numbers.
• The tree's structure ensures that each rational number appears only once and in its simplest terms, assisted by the gcd properties of the fractions.

By understanding how rational numbers are used within this structure, students can better grasp the efficiency and utility of the Calkin-Wilf tree.

Role of Mathematical Induction

Mathematical induction is a powerful proof technique used to establish the validity of a statement for all integers. This method is essential to prove the properties of the Calkin-Wilf tree, like the uniqueness of each fraction.

• Begin by proving the base case: The root \(\frac{1}{1}\) is correct.
• Assume the statement is true for a certain level in the tree.
• Then prove it must also be true for the next level based on the recursive rules \(\frac{i}{i+j}\) and \(\frac{i+j}{j}\).

Mathematical induction helps demonstrate the completeness and uniqueness of fraction representation in the tree, making it pivotal to understanding its construction and function.