Question

In the first quadrant of the \(x-y\) plane, draw a path that passes exactly once through cach point with both coordinates being integers. Each stopping place on the path should only be one unit right, one unit up, one unit left, or one unit down from the previous stopping place. Start the path at (0,0) . Use the path to construct a bijection from \(\mathrm{N}\) to \(\mathrm{N} \times \mathrm{N}\).

Short answer

Trace a zigzag path, assign numbers sequentially, and confirm it covers all points uniquely.

Step-by-step solution

Understand the Problem

We need to create a path starting from point (0,0) on the Cartesian plane such that it uniquely passes through each point where both coordinates are integers and all points are shifted by one unit in any direction adjacent (right, up, left, or down) from the previous point. This path will help us to construct a bijection from the set of natural numbers \( \mathbb{N} \to \mathbb{N} \times \mathbb{N} \).

Outline the Path

To trace the required path through each integer coordinate point, we will use a zigzag pattern starting from (0,0). The path goes right to (1,0), then up to (1,1), left to (0,1), and diagonally upward in a repetitive zigzag manner. For instance, from (0,0) move to (1,0) → (1,1) → (0,1) → (0,2) → (1,2) → (2,2) → (2,1) → (2,0) → (3,0) and so on. This process will be repeated to cover every lattice point.

Map the Path to Natural Numbers

Assign a unique natural number to each point along the path in the order they are traversed. Start with 1 for (0,0), 2 for (1,0), 3 for (1,1), and continue numbering each successive point on the path.

Verify the Bijection

Ensure that each natural number is assigned to exactly one unique pair of integers, and each pair of integers \( (m, n) \) in the first quadrant has a unique assigned natural number. We achieve this through the systematic zigzag traversal and unique sequential mapping.

Key concepts

Cartesian plane

The Cartesian plane is a fundamental concept in mathematics. It is a flat, two-dimensional surface that extends infinitely in all directions. You can think of it as a giant piece of graph paper. The plane is defined by two perpendicular axes. These axes are the x-axis (horizontal) and the y-axis (vertical). They intersect at a point called the origin, which has coordinates (0, 0).

Here are some key features of the Cartesian plane:

• **Coordinates**: Every point on this plane can be described using an ordered pair, (x, y), which represents its position along the x-axis and y-axis.
• **Quadrants**: The plane is divided into four quadrants. Each quadrant is formed by the positive and negative sides of the x and y-axes. The first quadrant, where this problem is set, is the top right quadrant, where both coordinates x and y are positive.
• **Grid Points**: On the Cartesian plane, grid points refer to points whose coordinates are integers, like (1, 2) or (3, 4).

The Cartesian plane provides a way to visualize and solve problems involving geometry and algebra. By creating paths on this plane, like in our exercise, we can establish connections between different sets of numbers, leading to deeper mathematical insights.

natural numbers

Natural numbers are the basic building blocks of arithmetic. They are simple and foundational, comprising the set of positive integers starting from 1 and continuing indefinitely: 1, 2, 3, 4, and so on. In some definitions, zero is also included as a natural number, though this can vary.

Some characteristics of natural numbers include:

• **Non-Negative**: Natural numbers are always greater than or equal to zero.
• **Whole Numbers**: They are whole numbers without any fractional or decimal part.
• **Infinite Set**: The set of natural numbers goes on forever, without end.

Natural numbers are often used in counting and ordering. In this exercise, they serve as identifiers, or labels, for different positions in a path on the Cartesian plane. This use connects spatial positions (integer coordinates on the plane) with sequential positions (natural numbers) through a bijection. By mapping natural numbers to grid points, we can systematically address every integer coordinate in the plane.

integer coordinates

Integer coordinates refer to positions on the Cartesian plane where both the x and y values are integers. These are specific points that sit precisely on the grid lines of the Cartesian plane, like (0, 0), (1, 2), or (3, 4). Each integer coordinate is a unique point that can be directly referenced in many mathematical contexts.

Important aspects of integer coordinates include:

• **Precision**: Integer coordinates are exact, making them ideal for constructing mathematical models or drawings.
• **Symmetry**: On the Cartesian plane, integer coordinates exhibit symmetrical properties based on the axes, beneficial in geometric constructions.
• **Usage in Bijections**: In the exercise, we use integer coordinates to establish a bijection with natural numbers. This bijection assigns a unique natural number to each integer coordinate in a path-covering process.

In the context of mathematical problem-solving, integer coordinates provide a framework for precise calculations and mappings. By understanding their arrangement and interactions, solutions like this bijection can be systematically constructed.