Problem 20 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The Cartesian product \(B \times A\) is given by the set \(\{(x, b), (x, c)\}\).

Step 1: Write down the given sets

Given sets are: \(A = \{b, c\}\), \(B = \{x\}\) and \(C = \{x,z\}\) We are to find the Cartesian product, \(B \times A\).

Step 2: Calculate the Cartesian product

To find the Cartesian product \(B \times A\), we need to find all ordered pairs \((a,b)\) such that \(a\) belongs to B and \(b\) belongs to A. B has only one element, which is \(x\), and A has two elements, \(b\) and \(c\). The ordered pairs are: \((x, b)\) and \((x, c)\)

Step 3: Write down the result

Now that we have found all ordered pairs for the Cartesian product \(B \times A\), we can write down the result as a set: $$B \times A = \{(x, b), (x, c)\}$$

Key Concepts

Ordered Pairs

In mathematics, an ordered pair is a crucial element when working with the Cartesian product. As the name suggests, an ordered pair consists of two elements arranged in a specific order. This order is significant because changing it can produce a different pair. For example, the ordered pair - $(a, b)$ is different from - $(b, a)$ unless both $a$ and $b$ are equal. Consider the ordered pair $(x, y)$; - The first element, $x$, is known as the first component or the first entry. - The second element, $y$, is the second component or entry. In the context of the Cartesian product, these ordered pairs are derived by taking one element from each of the involved sets to form every possible combination. This notion is pivotal in set theory as it helps to define relationships between different sets.

Set Theory

Set theory is the branch of mathematical logic that studies sets, which are collections of objects. Sets are fundamentals in creating more complex structures, like functions and orders, that form the backbone of mathematics. Here's a quick overview of how sets function and some of their properties:- **Elements of Sets**: The objects inside a set are called elements. - **Notation**: Sets are usually denoted by curly braces, like \(A = \{ b, c \}\). - **Subsets**: If every element of a set \(A\) is also in a set \(B\), then \(A\) is a subset of \(B\). In set theory, the concept of the Cartesian product is a formal way of combining two sets to form an entirely new set. Given the sets \(A\) and \(B\), the Cartesian product \(A \times B\) consists of all possible ordered pairs \((a, b)\) where \(a\) is an element of set \(A\) and \(b\) is an element of set \(B\).Set theory provides the tools needed to express and perform operations like union, intersection, difference, and Cartesian products in a systematic way.

Discrete Mathematics

Discrete mathematics involves the study of mathematical structures that are fundamentally discrete rather than continuous. It includes a variety of topics that do not require the notion of continuity. Here, we focus on the application of discrete structures in forming ordered pairs as part of the Cartesian product.In discrete mathematics: - We primarily work with sets, sequences, graphs, and integers. - Properties such as countability, distinctness, and separability are often emphasized.The formation of Cartesian products is a key operation in discrete mathematics. When you take two sets (consider the sets \(B=\{x\}\) and \(A=\{b, c\}\)), you apply discrete principles to combine these finite elements in a structured, pairwise manner. This approach is essential for the development of databases, algorithms, and systems where order and structure of data are crucial, as each entry can be distinctly categorized and accessed using this method.