Question

Sketch the sets \(X=[-1,3] \times[0,2]\) and \(Y=[0,3] \times[1,4]\) on the plane \(\mathbb{R}^{2}\). On separate drawings, shade in the sets \(X \cup Y, X \cap Y, X-Y\) and \(Y-X\).

Short answer

X union Y is the two rectangles representing X and Y drawn together. X intersect Y is a rectangle with corners at (0,1), (3,1), (0,2), (3,2). X minus Y is formed by two rectangles, with corners at (-1,0), (3,0), (-1,1), (3,1) and (0,2), (3,2), (0,1), (3,1). Y minus X is a rectangle with corners at (0,2), (3,2), (0,4), (3,4).

Step-by-step solution

Sketch the sets X and Y

Set X is a Cartesian product of intervals [-1,3] and [0,2]. This will form a rectangle with corners at points (-1,0), (3,0), (-1,2) and (3,2). Similarly, set Y is a Cartesian product of intervals [0,3] and [1,4], forming a rectangle with corners at points (0,1), (3,1), (0,4) and (3,4).

Sketch X union Y

The union of two sets is everything that is in either of the sets. On a diagram, this can be represented as both rectangles, including their interiors, sketched together on the same plane.

Sketch X intersect Y

The intersection of two sets is everything that is in both of the sets. On a diagram, it is the area where the two rectangles overlap. This region is a rectangle with corners at points (0,1), (3,1), (0,2) and (3,2).

Sketch X minus Y

The set difference X-Y is all the elements that are in X but not in Y. In the sketch, this is the part of the rectangle representing X that does not overlap with Y. It forms two rectangles with corners at points (-1,0), (3,0), (-1,1), (3,1) and (0,2), (3,2), (0,1), (3,1).

Sketch Y minus X

The set difference Y-X is all the elements that are in Y but not in X. In the sketch, this is the part of the rectangle representing Y that does not overlap with X. It is a rectangle with corners at points (0,2), (3,2), (0,4), (3,4).

Key concepts

Set Theory

Set theory is the mathematical study of collections of objects, known as sets. Any type of information can be bundled into a set, allowing mathematicians to deal with an assortment of elements as a single entity. A set is defined by its elements and the principle of extensionality, which states that two sets are equal if they have the same elements.

A fundamental operation in set theory is the Cartesian product, denoted as \times. If you have two sets, A and B, the Cartesian product is the set of all ordered pairs \big \(a, b\big \) where \(a \in A\) and \(b \in B\). For example, if set A has elements {1,2} and set B has elements {3,4}, the Cartesian product of A and B is \( \big\big \{ (1,3), (1,4), (2,3), (2,4) \big\big \} \). This concept allows for the creation of multi-dimensional sets, like the ones in the exercise, which can then be represented graphically as shapes on a plane.

Union and Intersection of Sets

In set theory, the concepts of union and intersection are crucial for understanding relationships between sets. The union of two sets, denoted as \(A \cup B\), is a set containing all elements that are either in A, in B, or in both. For sets represented as areas on the plane, the union would consist of the entire space covered by both sets.

The intersection of sets, denoted as \( A \cap B \), is a set containing only elements common to both A and B. If A and B are overlapping regions on the plane, their intersection is the overlapping area. Using our sets from the exercise, \(X \cup Y\) represents the entire area covered by both rectangles, while \(X \cap Y\) is the area where the two rectangles overlap, forming a new, smaller rectangle.

Set Difference

The set difference between two sets A and B, written as \( A - B \), or \( A \setminus B \), contains all elements that are in A but not in B. Unlike the union or intersection, this operation is not commutative; \( A - B \eq B - A \).

Take the exercise's example, where we find the set difference \( X - Y \). This is the part of set X that does not overlap with set Y, resulting in a certain area of the rectangle exclusive to X. Conversely, \( Y - X \), is the part of set Y that does not overlap with set X. It's an important concept that illustrates how we can carve out distinct regions between sets, adding to the richness of the relationships we can represent in set theory.