Question

Let \(X=\\{2,4 \mid, Y=\\{1,4 \mid,\) and \(Z=\\{0,4,8 \mid\). Construct the following sets: (a) \(X \times Y\) (b) \(X \times Y \times Z\) (c) \(Y \times Z\) (d) \(Z \times Y \times X\) (c) \(Z \times X \times Y\)

Short answer

Cartesian products for (a) X × Y, (b) X × Y × Z, (c) Y × Z, (d) Z × Y × X, and (e) Z × X × Y are combinations of set elements.

Step-by-step solution

Understand Cartesian Product

The Cartesian Product of two sets, say Set A and Set B, denoted as \(A \times B\), is a set consisting of all possible ordered pairs \((a, b)\) where \(a\) is in Set A and \(b\) is in Set B.

Calculate X × Y

Given sets \(X = \{2,4\}\) and \(Y = \{1,4\}\), the Cartesian product \(X \times Y\) consists of pairs \((x, y)\) where \(x\) is from \(X\) and \(y\) is from \(Y\). The pairs are: \((2, 1), (2, 4), (4, 1), (4, 4)\).

Calculate X × Y × Z

To calculate \(X \times Y \times Z\) with \(X = \{2,4\}\), \(Y = \{1,4\}\), and \(Z = \{0,4,8\}\), we combine the elements to form triplets of the form \((x, y, z)\). There are total \(|X| \times |Y| \times |Z| = 2 \times 2 \times 3 = 12\) elements. The full set is: \((2,1,0), (2,1,4), (2,1,8), (2,4,0), (2,4,4), (2,4,8), (4,1,0), (4,1,4), (4,1,8), (4,4,0), (4,4,4), (4,4,8)\).

Calculate Y × Z

For \(Y \times Z\) with \(Y = \{1,4\}\) and \(Z = \{0,4,8\}\), the set consists of pairs \((y, z)\): \((1, 0), (1, 4), (1, 8), (4, 0), (4, 4), (4, 8)\).

Calculate Z × Y × X

For \(Z \times Y \times X\), we form triplets \((z, y, x)\) from \(Z = \{0,4,8\}\), \(Y = \{1,4\}\), and \(X = \{2,4\}\). Similar to previous calculations, the entire set is: \((0,1,2), (0,1,4), (0,4,2), (0,4,4), (4,1,2), (4,1,4), (4,4,2), (4,4,4), (8,1,2), (8,1,4), (8,4,2), (8,4,4)\).

Calculate Z × X × Y

For \(Z \times X \times Y\), form triplets \((z, x, y)\) from elements in \(Z = \{0,4,8\}\), \(X = \{2,4\}\), and \(Y = \{1,4\}\). Resulting set will be \((0,2,1), (0,2,4), (0,4,1), (0,4,4), (4,2,1), (4,2,4), (4,4,1), (4,4,4), (8,2,1), (8,2,4), (8,4,1), (8,4,4)\).

Key concepts

Set Theory

Set Theory is a fundamental part of mathematics that allows us to study collections of objects. These objects can be numbers, letters, or pretty much anything. Sets are used to group related items together through unique elements.

Some important concepts in Set Theory include:

• Elements: These are individual objects within a set. For example, in the set \( X = \{2, 4\} \), 2 and 4 are elements.
• Subset: A set that contains some or all elements of another set. For example, \( \{2\} \) is a subset of \( X \).
• Union: The set containing all elements from two or more sets.
• Intersection: A set containing only the elements found in all sets being considered.

Understanding these basic principles helps when dealing with more complex problems such as finding Cartesian Products.

Ordered Pairs

Ordered pairs are a way to combine elements from two different sets. If you take Set A and Set B, the pair is often written as \((a, b)\), where \(a\) is from Set A and \(b\) is from Set B.

Here’s what to know about ordered pairs:

• Sequence Matters: In an ordered pair \((a, b)\), \(a\) always comes first and \(b\) second. Changing the order creates a different pair.
• Cartesian Product: The Cartesian Product of two sets is the complete set of ordered pairs you can create from those sets.
• Example: For \(X = \{2, 4\}\) and \(Y = \{1, 4\}\), the ordered pairs forming the Cartesian Product \(X \times Y\) include \((2, 1), (2, 4), (4, 1), (4, 4)\).

Using ordered pairs helps us visualize and work with combinations of elements from different sets.

Discrete Mathematics

Discrete Mathematics deals with structures that are fundamentally discrete rather than continuous. It covers a wide array of topics such as logic, algorithms, and of course, set theory.

Why is it important?

• Structure: Discrete math helps us understand the structure and relationship within sets, including the use of ordered pairs and Cartesian Products.
• Applications: It's used extensively in computer science for algorithms.
• Foundation: Discrete math lays the groundwork for various areas of mathematics, such as combinatorics and graph theory.

The study of these concepts provides a critical foundation for understanding complex systems in computer science and mathematics.

Mathematical Sets

Mathematical sets are collections of distinct objects or elements that share a common property. Sets are the building blocks of more complex mathematical structures and are essential in fields ranging from probability to geometry.

Here's what you need to know:

• Notation: Sets are usually denoted using curly braces, such as \(X = \{2, 4\}\).
• Types of Sets: Including finite sets (with limited elements) and infinite sets.
• Operations: You can perform various operations on sets like finding their union, intersection, or Cartesian Product.

In our example, the sets \(X, Y,\) and \(Z\) are finite, making it easier to compute their Cartesian Products. Understanding sets and their properties is crucial for mastering discrete mathematics.