Problem 21 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The Cartesian product of set A and the empty set is the empty set: \(A × ∅ = ∅\).

Step 1: Understand the given sets

We are given three sets A, B, and C. Let's represent them: \( A = \{b, c\} , B = \{x\}, C = \{x, z\} \)

Step 2: Calculate the Cartesian product of A and the empty set

We need to find the Cartesian product \(A × ∅\). Recall that the Cartesian product of any set and the empty set is always an empty set. \[ A × ∅ = \{(a, b) | a ∈ A, b ∈ ∅ \} \] Since there's no element in the empty set, there are no possible ordered pairs (a,b) that could be formed, so: \[ A × ∅ = ∅ \]

Step 3: Final Answer

The Cartesian product of set A and the empty set is the empty set: \[ A × ∅ = ∅ \]

Key Concepts

empty set

In mathematics, the empty set, denoted by \(\emptyset\), is a set that contains no elements. It is a unique and fundamental concept in set theory. No matter how hard you look inside it, you will find nothing. This might seem like a simple concept, but it's incredibly important.

Here are a few properties of the empty set that you should keep in mind:

• The empty set is a subset of every set. This means if you have a set \(A\), you can always consider \(\emptyset\) as a part of it.
• The empty set has exactly one subset, which is itself.
• The intersection of the empty set with any set is always the empty set. This makes intuitive sense because the empty set contains no elements to contribute to an intersection.

When we talk about operations like the Cartesian product, multiplying by the empty set means you'll get an empty set as a result. This is because there are no elements to pair with anything from another set, exactly as we've seen in the original solution for \(A \times \emptyset = \emptyset\).

ordered pairs

Ordered pairs are a basic building block in mathematics for describing relationships between two elements. An ordered pair consists of two elements grouped together in a specific order, usually written as \((a, b)\). The order of these elements matters; \((a, b)\) is not the same as \((b, a)\) unless \(a\) and \(b\) are the same.

Ordered pairs are crucial in defining Cartesian products. When you create a Cartesian product of two sets, you are essentially listing all possible ordered pairs that can be formed where the first element is from the first set and the second element is from the second set.

In the context of set theory, ordered pairs help us to understand relationships and mappings between different sets. They form the foundation for functions, which are special types of relations where each element in one set is related to exactly one element in another set.

set theory

Set theory is the branch of mathematical logic that studies sets, which are collections of objects. These objects can be anything: numbers, people, letters, or even other sets. Understanding sets is fundamental to mathematics because they form the basis for various structures and spaces in math.

In set theory:

• Elements are the objects contained within a set. For example, in the set \(A = \{b, c\}\), \(b\) and \(c\) are elements.
• We use special notations like \(\subseteq\) to denote subsets. A subset is a set that is entirely contained within another set.
• Operations like union, intersection, and Cartesian product are ways to combine or relate sets. For example, the Cartesian product \(A \times B\) combines sets \(A\) and \(B\) to form ordered pairs.

Set theory not only helps in understanding mathematical proofs but also provides the foundation for more complex topics, like functions and probability, by explaining how sets can relate to each other and be manipulated.