Problem 20 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The Cartesian product B x A is \(B \times A = \{(x, b), (x, c)\}\).

Step 1: Understand the Task

We are given sets A, B, and C, and our task is to find the Cartesian product of sets B and A, written as B x A.

Step 2: Write down the given sets

The given sets are: - A = {b, c} - B = {x} - C = {x, z}

Step 3: Identify the elements in the sets

Now, let's identify the elements in each set. For set A, we have: - b - c For set B, we have: - x

Step 4: Calculate the Cartesian product of B x A

Now that we know the elements of sets A and B, let's find the Cartesian product B x A. This means we will create ordered pairs where the first element comes from set B and the second element comes from set A. The Cartesian product B x A can be calculated as follows: B x A = {(x, b), (x, c)}

Step 5: Write the final answer

We have successfully calculated the Cartesian product B x A as: B x A = {(x, b), (x, c)}

Key Concepts

Set Theory

Set theory is the mathematical study of collections of objects, known as sets. In this context, a set is a well-defined collection of distinct objects, which could be anything: numbers, letters, or even other sets. Sets are usually denoted by capital letters such as \( A, B, \) or \( C \), and their elements are enclosed in curly braces, like \( \{a, b, c\} \).

• Sets can be finite or infinite, depending on whether they have a limited or unlimited number of elements.
• They can also be empty, containing no elements at all, called the null set or empty set, denoted by \( \emptyset \).
• In our exercise, we have sets \( A = \{b, c\} \), \( B = \{x\} \), and \( C = \{x, z\} \).

Set theory is foundational to many mathematical concepts, providing a unified way to describe various mathematical structures and operations.

Ordered Pairs

The concept of an ordered pair is central to understanding Cartesian products. An ordered pair is a pair of elements written in a specific sequence, usually denoted by parentheses \((a, b)\). Here, \( a \) is the first element, and \( b \) is the second element.

• The order in an ordered pair definitely matters, making \((a, b)\) different from \((b, a)\).
• Ordered pairs are used to represent elements of a Cartesian product, which is the set of all possible ordered pairs from two sets.

In our solution, we calculated the Cartesian product of sets \( B \) and \( A \) as \( B \times A = \{(x, b), (x, c)\} \). This set consists of ordered pairs where the first element is taken from set \( B \) and the second from set \( A \). Understanding ordered pairs helps clarify this arrangement, emphasizing the importance of sequence in these mathematical representations.

Discrete Mathematics

Discrete mathematics is a branch of mathematics dealing with countable, distinct elements. It includes a wide variety of topics such as graph theory, number theory, and importantly, set theory and relations, which are applicable to our exercise.

• Discrete mathematics contrasts with continuous mathematics, which deals with smoothly varying numbers and shapes.
• It is essential in computer science and operations research, especially in modeling relationships and processes electronically.
• Understanding discrete math helps us appreciate how elements from different sets can combine to form complex structures.

By studying Cartesian products, we delve into a core concept of discrete mathematics. This operation helps in constructing relations and functions, providing a clear and structured way to combine elements from multiple sets into new formations, such as ordered pairs. This is crucial in areas like databases, logic, and algorithm design.