Problem 22 - Discrete Mathematics with Applications

Question

Let \(A=\\{\mathrm{b}, \mathrm{c}\\}, B=\\{\mathrm{x}\\},\) and \(C=\\{\mathrm{x}, \mathrm{z}\\} .\) Find each set. $$A \times B \times \emptyset$$

Step-by-Step Solution

Answer

The Cartesian product of the three sets \(A, B\), and \(\emptyset\) is \(A \times B \times \emptyset = \emptyset\).

Step 1: Definition of Cartesian Product

The Cartesian product of two sets A and B, denoted by AxB, is the set of all possible ordered pairs with the first element from set A and the second element from set B. In other words, \(A \times B =\\{(a, b) | a \in A, b \in B\\}\). Now, we want to find the Cartesian product AxBx(empty set). Let's break it down step by step.

Step 1: Calculate AxB

We first need to calculate AxB, which is the Cartesian product of sets A and B. Given the sets \(A = \\{\mathrm{b}, \mathrm{c}\\}\) and \(B = \\{\mathrm{x}\\}\), we have: \(A \times B = \\{(a, b) | a \in A, b \in B\\} = \\{(b, x), (c, x)\\}\), since these are all the possible ordered pairs with the first element from A and the second element from B.

Step 2: Calculate (AxB)x(empty set)

Now, we'll calculate the Cartesian product of the set AxB and the empty set. The empty set, denoted by ∅, has no elements. Therefore, when finding the Cartesian product with an empty set, there will be no possible ordered pairs, as there are no elements in the empty set to form pairs. So: \((A \times B) \times \emptyset = \\{(a, b, c) | a \in A \times B, c \in \emptyset\\} = \emptyset\), as there are no elements in the empty set to form ordered triples. Therefore, the Cartesian product of three sets A, B, and ∅ is \(A \times B \times \emptyset = \emptyset\).

Key Concepts

Understanding Set Theory

Set theory is a crucial foundation of mathematics. It deals with the study of sets, which are collections of distinct objects. These objects, called elements, could be numbers, letters, or any collection of things.
Understanding basic operations like unions, intersections, and Cartesian products helps us grasp more complex mathematical ideas.

• **Set Notation**: Sets are usually represented by capital letters, and elements are enclosed in curly brackets, like \( A = \{a, b, c\} \).
• **Operations**: You can perform operations like union (combining elements), intersection (common elements), and Cartesian products (pairs or lists of elements) on sets.

Essentially, set theory provides a structured way of describing and analyzing the relationships between different collections.

Exploring the Empty Set

The empty set, denoted as \( \emptyset \), is a unique set that contains no elements. It's a fundamental concept in set theory and forms the basis for understanding more complicated constructs.
Since it has no elements, any operation involving an empty set results in a predictable outcome.

• **Properties**: The cardinality (size) of the empty set is zero, because it contains no elements.
• **Interactions**: Any set union with an empty set results in the original set, while any intersection yields the empty set.

In Cartesian products, multiplying by the empty set always results in \( \emptyset \), as there are no elements to combine.

Defining Ordered Pairs

Ordered pairs are fundamental to the concept of Cartesian products. An ordered pair \( (a, b) \) consists of two elements where the sequence is important.
This order distinguishes them from sets where order doesn't matter.

• **Notation**: Ordered pairs are represented as \( (a, b) \), where \( a \) and \( b \) are elements from respective sets.
• **Application**: Used in coordinate geometry and functions, ordered pairs provide a structured way to describe relationships.

In our problem, the ordered pairs \( (b, x) \) and \( (c, x) \) were formed from sets \( A \) and \( B \).

Understanding Ordered Triples

Much like ordered pairs, ordered triples extend the concept to three elements. In ordered triples \( (a, b, c) \), order is crucial, and each element comes from a different set.
Again, these highlight how Cartesian products can be extended to multiple dimensions.

• **Notation**: They are written as \( (a, b, c) \).
• **Role in Cartesian Products**: While ordered pairs consist of two elements, ordered triples involve three, expanding their utility in three-dimensional spaces.

In the original exercise, involving a triple with an empty set results in an empty set, showing the constraints of working with elements that do not exist.