Question

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

Short answer

For \(X \cup Y\), the shaded region includes the entire circle and the rectangular area below the x-axis and outside the circle. For \(X \cap Y\), the shaded region takes on a semicircular shape, representing the lower half of the circle. For \(X-Y\), we hold the upper half of the circle, while \(Y-X\) gives a rectangular region below the x-axis and outside the circle.

Step-by-step solution

Sketch Set X and Set Y

For set X, we are given a circle's equation centered at the origin (0,0) with radius 1, including both the interior and boundary of the circle due to the \(\leq\) sign in the inequality. Draw this circle on the plane. For set Y, the inequality \(-1 \leq y \leq 0\) corresponds to a horizontal strip of points between y = -1 and y = 0, including the lines y = -1 and y = 0.

Sketch Set \(X \cup Y\)

The union of two sets is the set of elements which are in X, in Y, or in both. So, to sketch \(X \cup Y\), simply combine both shaded regions of X and Y on the plane.

Sketch Set \(X \cap Y\)

The intersection of two sets is the set of elements which are in both X and Y. So, to sketch \(X \cap Y\), shade in only the region where the shaded areas of X and Y overlap.

Sketch Set \(X-Y\) and \(Y-X\)

The difference between two sets X and Y is the set of elements which are in X but not in Y. For \(X-Y\), therefore, we need to remove from X everything that is in Y, which means removing the part of the circle below the x-axis (the region of overlap with Y). To sketch \(Y-X\), we need to remove from Y everything that falls inside the circle, leaving only the rectangular region below the x-axis and outside the circle.

Key concepts

Sketching Sets in R^2

When visualizing sets, especially in two dimensions (R^2), the use of a coordinate system is essential. It helps to represent and understand the position and extension of sets within a plane.

Set X in our exercise is a circle with a radius of 1, centered at the origin. To sketch this, you'll draw a round shape where every point on the perimeter is exactly 1 unit away from the center (0,0), and make sure to include the area within the circle. This represents all the points (x, y) that satisfy the equation given.

Set Y, on the other hand, is a horizontal strip going from y = -1 to y = 0. To represent this, draw two parallel lines horizontally across the plane from y = -1 to y = 0 and shade the area in between to illustrate that all the points along this strip are included in Set Y. A correct sketch aids in the visualization of complicated concepts and renders subsequent operations, such as finding unions and intersections, more straightforward.

Union of Sets

The union of sets combines all the elements from the given sets. In terms of sketching, this is represented by shading the areas covered by both sets, without leaving any parts out.

Imagine layering transparent sheets on top of one another; every region covered by at least one sheet gets shaded. So, when you look at the exercise for the union of sets X and Y, \(X \cup Y\), your drawing would show the entire circle and the entire strip, with an overlapping region where both sets coincide on the plane. This concept is pivotal in analyzing the collective behavior of two datasets in a shared space.

Intersection of Sets

Conversely, the intersection of sets narrows down to the elements they share. This is similar to seeking out the common ground between two groups. Visually, it's where their shaded regions overlap.

In our example, \(X \cap Y\), the intersection of the circle and the strip, is the crescent-shaped area at the bottom of the circle. Here, you would shade in only the section where both the circle and the horizontal strip sit atop each other. Understanding intersections is invaluable when identifying common characteristics between different sets or systems.

Set Difference

The set difference might be less intuitive at first glance. Think of it as taking away the shared elements from one set. If we have two sets, X and Y, the difference \(X - Y\) includes only those elements that are in X but not in Y.

To sketch this, shade the entire area of Set X and then 'erase' or exclude the region where it overlaps with Set Y. For \(X - Y\), this leaves us with the top semicircle. Conversely, \(Y - X\) shows us the part of the strip that does not touch the circle at all. This operation is similar to filtering out common attributes to focus on what makes one set distinct from another.