Question

Let \(A=\\{0,3\\}\) and \(B=\\{x, y, z\\}\). Find the following: (a) \(A \times B\) (b) \(A \times A \times B\) (c) \(B \times A\) (d) \(B \times A \times B\)

Short answer

(a) \(A \times B = \{(0, x), (0, y), (0, z), (3, x), (3, y), (3, z)\}\); (b) \(A \times A \times B = \{(0,0,x), (0,0,y), (0,0,z), (0,3,x), (0,3,y), (0,3,z), (3,0,x), (3,0,y), (3,0,z), (3,3,x), (3,3,y), (3,3,z)\}\); (c) \(B \times A = \{(x,0), (x,3), (y,0), (y,3), (z,0), (z,3)\}\); (d) \(B \times A \times B = \{(x,0,x), (x,0,y), (x,0,z), (x,3,x), (x,3,y), (x,3,z), (y,0,x), (y,0,y), (y,0,z), (y,3,x), (y,3,y), (y,3,z), (z,0,x), (z,0,y), (z,0,z), (z,3,x), (z,3,y), (z,3,z)\}\).

Step-by-step solution

Understanding Cartesian Product

The Cartesian product of two sets \(A\) and \(B\) is the set consisting of all possible ordered pairs \((a, b)\) such that \(a \in A\) and \(b \in B\). For example, if \(A = \{0, 3\}\) and \(B = \{x, y, z\}\), then \(A \times B = \{(0, x), (0, y), (0, z), (3, x), (3, y), (3, z)\}\).

Calculating A × B

Based on the definition of Cartesian product, we need to form pairs by combining each element of \(A\) with each element of \(B\). \ \(A = \{0, 3\}\), \(B = \{x, y, z\}\). Thus, \ \(A \times B = \{(0, x), (0, y), (0, z), (3, x), (3, y), (3, z)\}\).

Calculating A × A × B

For \(A \times A \times B\), we form ordered triples \((a_1, a_2, b)\) where \(a_1, a_2 \in A\) and \(b \in B\). Each element \(a_1\) from \(A\) can be paired with each element \(a_2\) from \(A\), and then paired with each element from \(B\). \ \(A \times A \times B = \{(0, 0, x), (0, 0, y), (0, 0, z), (0, 3, x), (0, 3, y), (0, 3, z), (3, 0, x), (3, 0, y), (3, 0, z), (3, 3, x), (3, 3, y), (3, 3, z)\}\).

Calculating B × A

According to the definition of Cartesian product, we form pairs by combining each element of \(B\) with each element of \(A\). \ \(B \times A = \{(x, 0), (x, 3), (y, 0), (y, 3), (z, 0), (z, 3)\}\).

Calculating B × A × B

For \(B \times A \times B\), we form ordered triples \((b_1, a, b_2)\) where \(b_1, b_2 \in B\) and \(a \in A\). Each element \(b_1\) from \(B\) can be paired with each element \(a\) from \(A\), and then paired with each element from \(B\) again. \ \(B \times A \times B = \{(x, 0, x), (x, 0, y), (x, 0, z), (x, 3, x), (x, 3, y), (x, 3, z), (y, 0, x), (y, 0, y), (y, 0, z), (y, 3, x), (y, 3, y), (y, 3, z), (z, 0, x), (z, 0, y), (z, 0, z), (z, 3, x), (z, 3, y), (z, 3, z)\}\).

Key concepts

Discrete Mathematics

Discrete Mathematics is a branch of mathematics dealing with objects that can assume only distinct, separate values. This field is crucial in computer science, cryptography, and combinatorics because it deals with for example integers, graphs, and logical statements. Unlike continuous mathematics, which concerns smooth changes and real numbers, discrete mathematics focuses on countable, often finite, sets and structures.
One of the core areas in discrete mathematics is the study of sets and their interactions. The sets' operations, such as union, intersection, and Cartesian products, enable us to illustrate and solve problems systematically. By working with sets and understanding their properties, you can handle complicated theoretical questions that have practical applications in the digital world.
For instance, the Cartesian product discussed here is a way of constructing new sets by combining existing ones. This topic is vital in designing algorithms where organizing data into structured pairs or triples is necessary.

Ordered Pairs

Ordered pairs are foundational in set theory, representing elements with a specific order. An ordered pair takes the form \((a, b)\), where \(a\) is known as the first component and \(b\) is the second component. This principle is used widely in Cartesian products, where each combination of elements is arranged into a pair.
Unlike simple sets, where the order doesn't matter, an ordered pair relies on this specific order for its identity. For example, considering the sets \(A = \{0, 3\}\) and \(B = \{x, y, z\}\), forming the Cartesian product \(A \times B\) gives ordered pairs like \((0, x)\) and \((3, y)\). Each pair has a distinct first element from \(A\) and a second element from \(B\).
This structure is essential when discussing coordinates in mathematics, like in the Cartesian plane, enabling precise data representation and manipulation.

Set Theory

Set Theory is a fundamental part of mathematics, focusing on the study of collections of objects, known as sets. In set theory, the concept of a set is considered a crucial primitive notion. It helps in understanding and establishing the foundation for various mathematical theories.
With set theory, we can define operations like union, intersection, and Cartesian product. Such operations allow us to comprehend how sets combine and relate to one another, forming complex structures from simple building blocks. In our example, sets \(A\) and \(B\) become intertwined through the operation of Cartesian product, unraveling new sets by pairing elements of one set with those of another.
This makes set theory not just an abstract mathematical concept but a vital tool in real-world applications too. It's used in database management, programming, and networking, where understanding relationships between different sets of information is key.

Combinatorics

Combinatorics is the area of mathematics that focuses on counting, arranging, and finding patterns or structures in sets. It's an essential field for solving problems related to the selection and arrangement of objects. The goal is to quantify possibilities and find efficient solutions to complex problems.
In regards to Cartesian products, combinatorics allows us to count and explore the number of possible ordered combinations that can be formed. By examining sets like \(A = \{0, 3\}\) and \(B = \{x, y, z\}\), combinatorics empowers us to calculate how many possible ordered pairs, triples, or larger multi-element groups we can create. For instance, forming \(A \times B\) results in 6 pairs, while \(A \times A \times B\) leads to 12 possible ordered triples.
Combinatorics aids in understanding the ways items can be organized and combined, making it a cornerstone in optimization, probability, and algorithm design.