Question

Sketch the sets \(X=[1,3] \times[1,3]\) and \(Y=[2,4] \times[2,4]\) on the plane \(\mathbb{R}^{2}\). On separate drawings, shade in the sets \(X \cup Y, X \cap Y, X-Y\) and \(Y-X .\) (Hint: \(X\) and \(Y\) are Cartesian products of intervals. You may wish to review how you drew sets like \([1,3] \times[1,3]\) in the exercises for Section 1.2.)

Short answer

The sketch of the sets X and Y are squares in the plane centered at (2,2) and (3,3) respectively. Their union is a larger square that includes all points from both sets. Their intersection is a square at the center where they overlap. The difference X-Y consists of two squares on the bottom left and top left of X not covered by Y while the difference Y-X consists of two squares on the top right and bottom right of Y not covered by X.

Step-by-step solution

Sketch the Set X

The set X is a Cartesian product of closed intervals [1, 3] and [1, 3]. This means that X includes all points in the plane that have both x-coordinate and y-coordinate between 1 and 3, inclusive. Sketch a square on the plane that has corners at points (1,1), (3,1), (1,3), and (3,3).

Sketch the Set Y

Y is a Cartesian product of closed intervals [2, 4] and [2, 4]. This means that Y includes all points in the plane where both x-coordinate and y-coordinate are between 2 and 4. On a new plot, sketch a square that has its corners at points (2,2), (4,2), (2,4), and (4,4).

Union of X and Y

The union of two sets combines all the elements from both sets. X ∪ Y includes all points that are in X or in Y or in both. On a new plot, shade in the squares that represent X and Y.

Intersection of X and Y

The intersection of two sets contains elements that are common to both sets. X ∩ Y includes all points that are in both X and Y. On a new plot, shade in the area where the squares that represent X and Y overlap. This will correspond to a square bounded by (2,2), (3,2), (2,3), and (3,3).

Difference X - Y

The set difference X – Y includes all points that are in X and not in Y. On a new plot, shade in the area of the square representing X that is not covered by the square representing Y. This will correspond to a square bounded by (1,1), (3,1), (1,2), and (3,2) and another square bounded by (1,3), (3,3), (1,2), and (3,2).

Difference Y - X

The set difference Y – X includes all points that are in Y and not in X. On a new plot, shade in the area of the square representing Y that is not covered by the square representing X. This will correspond to a square bounded by (2,4), (4,4), (2,3), and (4,3) and another square bounded by (4,3), (4,2), (3,2), and (3,3).

Key concepts

Set Theory

Set theory is the mathematical study of collections of objects, known as sets. In set theory, objects within a set are called elements or members, and they can be anything from numbers to letters to geometric shapes. The beauty of set theory lies in its ability to organize and categorize data into well-defined groups.

The concept of a set is fundamental in mathematics; it serves as the building block for more complex structures like sequences, functions, spaces, and more. Set theory also introduces several operations that can be performed on sets, such as unions, intersections, and set differences. These operations help us understand the relationships between sets and are particularly useful in fields like logic, probability, and topology.

For example, the Cartesian product, as seen in the original exercise, is a concept from set theory that combines sets to form pairs of elements, leading to a new set. This particular operation is crucial in defining coordinate systems and multidimensional spaces.

Union and Intersection of Sets

Understanding the union and intersection of sets is quite intuitive once you associate them with everyday language. The union of two sets, denoted as \(A \bigcup B\), is a set containing all the elements from both sets \(A\) and \(B\), without any duplicates. It's like bringing two groups together for a party and including everyone.

On the other hand, the intersection of two sets, denoted as \(A \bigcap B\), is where things get collaborative. It's the set of all elements that are common to both sets \(A\) and \(B\). It's like finding common friends between two separate friend circles.

In the context of our exercise with sets \(X\) and \(Y\), picturing these operations graphically as shaded areas on the plane helps us cement our understanding of how these sets interact in space. The union brings two squares together, while the intersection reveals their shared area.

Set Difference

When we talk about the set difference, it's like we're looking for what's unique in one set that the other doesn't possess. Mathematically, we denote the set difference of \(A\) and \(B\) as \(A - B\). This gives us a new set comprised of elements that are in set \(A\) but not in set \(B\).

For visual learners, the set difference can be seen graphically as part of the square that's not overlapped by another, akin to cutting out a piece of the picture. In the supplied example with sets \(X\) and \(Y\), the set difference \(X - Y\) is the region where \(X\) stands alone, and similarly, \(Y - X\) is where \(Y\) exists without \(X\). These operations are particularly noteworthy in probability and statistics to determine exclusive events.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface formed by the intersection of a vertical line (y-axis) and a horizontal line (x-axis). This plane is used to visually represent relationships between two variables, typically in the form of coordinates represented by ordered pairs (x, y).

On a coordinate plane, every point is unique and corresponds to a precise location. This becomes especially practical when dealing with Cartesian products of sets as in our example, as it allows us to draw squares representing the sets \(X\) and \(Y\) and visually interpret operations such as unions and intersections by simply identifying the relevant areas on the plane.

By enhancing the understanding of Cartesian planes, students not only improve their ability to work with geometric shapes on the plane but also lay the groundwork for advanced mathematics, including calculus and vector spaces.