Question

Sketch these Cartesian products on the \(x-y\) plane \(\mathbb{R}^{2}\) (or \(\mathbb{R}^{3}\) for the last two). $$ \mathbb{N} \times \mathbb{Z} $$

Short answer

The sketch of the Cartesian product \(\mathbb{N} \times \mathbb{Z}\) will be an infinite grid of points that fill the right half of the \(x-y\) plane including the origin. Each vertical line in the grid represents an integer from the set \(\mathbb{N}\), and on each line there are points corresponding to every integer from the set \(\mathbb{Z}\).

Step-by-step solution

Understanding the Sets and their Cartesian Product

Understand that \(\mathbb{N}\) represents the set of natural numbers (0, 1, 2, 3,...) and \(\mathbb{Z}\) represents the set of integers (...-3, -2, -1, 0, 1, 2, 3,...). The Cartesian product \(\mathbb{N} \times \mathbb{Z}\) would therefore give you ordered pairs where the first element are from \(\mathbb{N}\) and the second element from \(\mathbb{Z}\).

Plotting the Points

The points represented by \(\mathbb{N} \times \mathbb{Z}\) are (0,-3), (0,-2), (0,-1), (0,0), (0,1), (0,2), (0,3), (1,-3), (1,-2), (1,-1), (1,0), (1,1), (1,2), (1,3), and so on. Plot these points on the \(x-y\) plane, placing the first element of each ordered pair on the x-axis and the second one on the y-axis.

Visualizing the Result

The final sketch will be an infinite grid of points that fill up the right half (including origin) of the \(x-y\) plane because all the x-values (coming from \(\mathbb{N}\)) are 0 or positive, while y-values (coming from \(\mathbb{Z}\)) can be both negative and positive.

Key concepts

Set of Natural Numbers

The set of natural numbers, typically denoted by the symbol \( \mathbb{N} \), includes all the whole numbers starting from 0 (or 1, depending on the definition) extending to infinity in positive direction. This set is expressed as \( \mathbb{N} = \{0, 1, 2, 3, ...\} \). It's important to note that natural numbers do not include negative numbers or fractions.

Natural numbers are foundational in mathematics because they are used for counting and ordering. When you're working with sets, particularly in the context of Cartesian products, you're often dealing with an infinite number of pairings with other sets, which poses unique challenges for plotting or visualization.

For example, when we consider the Cartesian product with another set, like the set of integers, each natural number is paired with every integer to form a set of ordered pairs, creating a vast matrix of points when plotted on a coordinate plane.

Set of Integers

In contrast to the natural numbers, the set of integers, represented as \( \mathbb{Z} \), encompasses all whole numbers including the negatives. This set can be written as \( \mathbb{Z} = \{..., -3, -2, -1, 0, 1, 2, 3, ...\} \).

The inclusion of negative numbers expands the potential outcomes when combining the set of integers with other sets, such as natural numbers, in operations like the Cartesian product. The resultant paired elements give rise to a broad array of points.

Importance of Zero

The integer zero serves as a pivotal point—the bridge between the positive and negative elements within the set. Not only does it provide symmetry, but in Cartesian products, it often creates the origin point from which other points extend into the four quadrants of a coordinate system.

Plotting Points on the Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface defined by a horizontal axis (x-axis) and a vertical axis (y-axis). The intersection of these axes at point (0, 0) is known as the origin.

To plot points resulting from a Cartesian product such as \( \mathbb{N} \times \mathbb{Z} \), you identify each ordered pair's 'x' and 'y' values and find their corresponding location on the plane. For instance, in the ordered pair (1, -2), the number 1 is the 'x' value (from \( \mathbb{N} \) in this case), and -2 is the 'y' value (from \( \mathbb{Z} \)). Here's how to plot:

• Start at the origin (0, 0).
• Move right by 1 unit on the x-axis for the x value of 1.
• Then move down by 2 units on the y-axis for the y value of -2.
• Mark this point on the plane.

This process is repeated for each ordered pair in the Cartesian product. The result, particularly for products like \( \mathbb{N} \times \mathbb{Z} \), manifests as a lattice of points that cover one side or multiple quadrants of the plane, depending on the direction and magnitude of the sets involved.