Question

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

Short answer

The Cartesian product \(\mathbb{Z} \times \mathbb{Z}\) on the \(x-y\) plane forms an infinite grid with points at every intersection of the lines. Each point represents a pair of integers.

Step-by-step solution

Understanding the Coordinates

The Cartesian product \(\mathbb{Z} \times \mathbb{Z}\) represents all pairs of integers. Each pair of integers can be treated as an \((x, y)\) coordinate, where each of \(x\) and \(y\) are any possible integer.

Plotting the Points

Plot these points on the \(x-y\) plane by marking the coordinates that interpolate at integer points. Keep in mind that the points would be infinite in all directions since the set of integers is infinite.

Interpretation

While you can't mark all points on a paper, you should clearly understand that it's an infinite grid filling up the entire plane since any pair of integers can be found on the grid. The points are positioned at each intersection of the lines in the grid.

Key concepts

Understanding Coordinates

When we encounter the concept of coordinates, we're talking about a set of values that determine the exact position of a point in a space. In the context of a two-dimensional plane, such as the Cartesian plane, these values are typically written in the form \( (x, y) \) where \( x \) and \( y \) represent distinct values along the horizontal (x-axis) and vertical (y-axis) axes, respectively.

The beauty of coordinates is their universality; they allow us to pinpoint the location of any point in a precise manner that can be easily understood and communicated. When we say a point has coordinates \( (3, 4) \), we mean that the point is located three units to the right along the x-axis (since 3 is positive) and four units up along the y-axis (since 4 is positive) from where the two axes intersect, also known as the origin.

Integer Points on the Cartesian Plane

Integer points, as the name suggests, refer to coordinates on the Cartesian plane where both \( x \) and \( y \) are integers. Due to the nature of integers being whole numbers that include negative values, zero, and positive values, the integer points on the plane create a vast grid that extends infinitely in all directions.

The collection of all possible ordered pairs of integers \( (x, y) \) forms an infinite set known as \( \mathbb{Z} \times \mathbb{Z} \). Visualizing this on the Cartesian plane, you can think of it like a checkerboard that spans endlessly, with each square on the board representing an integer point. Each intersecting line along the x-axis and y-axis corresponds to an integer value, creating a matrix of dots that embody this concept of integer points. Remember, every single dot in this grid symbolizes an \( (x, y) \) coordinate where \( x \) and \( y \) are integers.

The x-y Plane

The \( x-y \) plane is the foundational stage where we visualize coordinates and understand concepts such as integer points and Cartesian products. Also known as the Cartesian plane in honor of the French philosopher and mathematician René Descartes, it consists of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is called the origin, typically labeled as \( (0, 0) \).

The x-axis represents all possible values of \( x \), from negative to positive infinity, and the y-axis does the same for \( y \). When we locate an integer pair \( (x, y) \) on this plane, we're placing a point at the cross-section of the 'x' value along the horizontal axis and the 'y' value along the vertical axis. The resulting grid, formed by lines drawn at each integer mark along the axes, serves as a guide for plotting points and understanding their relationships in the form of lines, curves, and shapes.