Question

Sketch the sets \(X=\left\\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2} \leq 1\right\\}\) and \(Y=\left\\{(x, y) \in \mathbb{R}^{2}: x \geq 0\right\\}\) on \(\mathbb{R}^{2}\). On separate drawings, shade in the sets \(X \cup Y, X \cap Y, X-Y\) and \(Y-X\).

Short answer

To solve this problem, each set needs to be sketched on \(\mathbb{R}^2\), keeping in mind the definitions for each. X is the disk centered at the origin with radius 1, Y is everything to the right of (and including) the y-axis. The union X∪Y is all of X and Y while the intersection X∩Y is the right half of the disk. X-Y corresponds to the left half of the disk, and Y-X corresponds to everything to the right of the y-axis but outside of the disk.

Step-by-step solution

Sketching the Sets X and Y

To sketch set X, you should note that it represents all the points (x, y) that satisfy the inequality \(x^2 + y^2 \leq 1\), which corresponds to all the points inside the circumference of radius 1, including the circumference itself. This is a disk centered at the origin with radius 1. To sketch set Y, note that it contains all the points \(x, y\) for which \(x \geq 0\), which corresponds to everything on and to the right of the Y-axis.

Sketching the Union and Intersection of X and Y

The union of X and Y, denoted by X∪Y, is the set containing all the elements of X and Y. To represent it, you need to shade in both areas represented by X and Y. The intersection of X and Y, denoted by X∩Y, is the set containing only the elements that belong to both X and Y. In the graph, this is represented by the area where the sets X and Y overlap, or, in other words, the right half of the disk.

Sketching the Difference of the Sets

Set X-Y, means all points in X that are not in Y, or the left half of the disk. Set Y-X means all points in Y that aren’t in X, or equivalently, everything to the right of the y-axis but outside the disk.