Problem 26 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Cartesian product of three sets

We have three sets, \(A\), \(B\), and \(C\), and we want to find the Cartesian product of these sets, which is denoted as \(A \times C \times B\). The Cartesian product is the set of all possible ordered triples \((a, c, b)\) with \(a \in A, c \in C,\) and \(b \in B\).

Step 2: List the elements of each set

First, we list the elements of each set to have a clear picture of the sets we are working with: - \(A = \{b, c\}\) - \(B = \{x\}\) - \(C = \{x, z\}\)

Step 3: Construct the ordered triples

Now, we find all possible ordered triples by selecting one element from each set in the correct order, as explained in Step 1. 1. For \(a = b\): - For \(c = x\): - For \(b = x\), we have an ordered triple \((b, x, x)\) - For \(c = z\): - For \(b = x\), we have an ordered triple \((b, z, x)\) 2. For \(a = c\): - For \(c = x\): - For \(b = x\), we have an ordered triple \((c, x, x)\) - For \(c = z\): - For \(b = x\), we have an ordered triple \((c, z, x)\)

Step 4: Write the final result as a set

Now we combine all the ordered triples to form the Cartesian product, \(A \times C \times B\): \[ A \times C \times B = \{(b, x, x), (b, z, x), (c, x, x), (c, z, x)\} \] Thus, the Cartesian product of the given sets is \(\{(b, x, x), (b, z, x), (c, x, x), (c, z, x)\}\).

Key Concepts

ordered triples

In the realm of set operations and discrete mathematics, an ordered triple is a fundamental concept that extends basic principles of order within sets. Imagine it as a trio of elements where the sequence matters. Unlike sets where the order is irrelevant, in ordered triples, each position has significance.

- The first element belongs to the first set. In our example, this is set A.- The second element belongs to the second set, which is set C in our case.- The third element belongs to the last set, here represented by set B.
When forming ordered triples, we combine these components in a specific sequence, \( (a, c, b) \), ensuring that we select one item from each set in the order specified. This is a critical part of understanding how Cartesian products function, as each ordered triple unites elements across multiple dimensions, maintaining the integrity of their arrangement.

set operations

Set operations form the backbone of combinatorial processes in discrete mathematics, providing the framework for constructing and analyzing collections of objects. One crucial operation is the Cartesian product, used here to create ordered triples.

The Cartesian product of three sets, for example, \( A \times C \times B \), results in a new set. This set consists of all possible combinations where the first element is taken from set A, the second from set C, and the third from set B.

• From \( A = \{b, c\} \)
• From \( C = \{x, z\} \)
• From \( B = \{x\} \)

Set operations, such as the Cartesian product, are instrumental in fields where relationships and logical data structuring are vital. They ensure systematic combination and provide a calculated approach to unraveling complex problems, whether in computer science algorithms, database structuring, or even permutations in probability.

discrete mathematics

Discrete mathematics is a branch of mathematics dealing with structures that are fundamentally distinct and separated, unlike continuous mathematics which deals with continuous processes. Within discrete mathematics, concepts like sets, ordered collections, and Cartesian products are pivotal because they form foundational tools for algorithms, cryptography, and information theory.

The study of set operations, such as the one illustrated with \( A \times C \times B \), allows students to delve into ways data can be grouped and ordered in logical sequences, facilitating problem-solving and theoretical proof development.

• It influences computation through logical reasoning.
• It's applied in advanced counting techniques and coding theory.
• And it supports algorithmic efficiency by clarifying the structure of data relationships.

Ultimately, through ordered triples and set operations, discrete mathematics unlocks structured ways of looking at a world full of categorical and finite elements, providing the tools necessary to analyze practical and theoretical problems alike.