Question

Sketch the following sets of points in the \(x-y\) plane. $$ \left\\{(x, y): x, y \in \mathbb{R}, x^{2}+y^{2} \leq 1\right\\} $$

Short answer

The set of points that satisfy the inequality \(x^{2}+y^{2} \leq 1\) represents a filled circle, i.e., a disc, of radius 1 centered at the origin in the x-y plane. It includes all points within the boundary of the circle and the points on the boundary of the circle.

Step-by-step solution

Identify the equation

Analyzing the equation given, \((x, y): x, y \in \mathbb{R}, x^{2}+y^{2} \leq 1\), we can recognize this as the analytical description of a disc. In a two dimensional Cartesian coordinate system, the equation \(x^{2}+y^{2} = r^{2}\) represents a circle with radius r centered at the origin (0,0). The equation \(x^{2}+y^{2} \leq r^{2}\), however, represents a disk with radius r, also centered at the origin. Thus, in our case, the set describes a disc with a radius of 1 centered at the origin.

Coordinate identification

The equation \(x^{2}+y^{2} \leq 1\) indicates a disc of radius 1 centered at the origin (0,0). So, the all values of x and y that satisfy this inequality will be the coordinates of the points within the disc and on the boundary of the disc (the circle of radius 1).

Graph plotting

To plot the graph, draw a circle of radius 1 using the origin (0,0) as the center. Then, fill the region of the circle, including the boundary, as it represents all the points that satisfy the inequality \(x^{2}+y^{2} \leq 1\).

Key concepts

Coordinate System

A coordinate system is essentially the backdrop for mapping points, lines, and shapes in mathematics. It's a way of assigning a set of numbers, or coordinates, to every point in a plane that uniquely identifies its location.

One of the most common systems used is the Cartesian coordinate system, also known as a rectangular coordinate system. It’s a two-dimensional system defined by a pair of perpendicular axes: the horizontal axis, typically called the x-axis, and the vertical axis, known as y-axis.

The Basics of Coordinates

Each point on this plane can be described by an ordered pair of numbers (x, y). The first number, x, represents the point's horizontal position relative to the origin (0,0), which is the point where the two axes intersect. The second number, y, indicates the vertical position. By using this system, any location within the plane can be pinpointed accurately.

In the context of the exercise, the students are tasked to determine the points that lie within a disc on this plane, making the concept of a coordinate system crucial to understanding and solving the problem.

Cartesian Plane

The Cartesian plane is named after René Descartes, who laid the foundation for analytic geometry. It's a two-dimensional surface formed by the intersection of two number lines, the x-axis and y-axis, at a right angle to each other, creating four quadrants. Each quadrant is a unique combination of positive and negative values for x and y.

Quadrants and Points

For instance, in the first quadrant, both x and y are positive, while in the second, x is negative and y is positive, and so on. This helps in identifying the position of points and interpreting equations graphically.

The Cartesian plane is an invaluable tool for visualizing and solving equations. For the exercise provided, students need to place the concept of a disc on the plane. The plane makes it easier to see the set of points that meet the condition of the equation, helping students grasp the concept of regions defined by inequalities.

Inequalities in Algebra

Inequalities are a fundamental part of algebra that express the relationship between two values when they are not strictly equal. They are represented by symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Inequalities can define intervals, ranges, or regions within the coordinate system where a set of conditions hold true.

Graphing Inequalities

When dealing with inequalities in a two-dimensional context, they describe not just lines but entire areas on the Cartesian plane. The exercise involving the disc helps visualize a two-variable inequality. Here, rather than a line, the inequality describes a circle and all the points inside it—the area is the solution to the inequality.

Understanding how to graph these inequalities helps students solve and visualize many types of algebraic problems, particularly those involving areas or regions, like the one in the exercise.