Problem 35 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The short version of the answer is: \(A \times (B \cup C) = \\{(b,x), (b,z), (c,x), (c,z)\\}\)

Step 1: Find the Union of Sets B and C

First, we need to find the union of sets B and C. The union of two sets contains all the unique elements from both sets. We have B = {x} and C = {x, z}. The union B ∪ C is: \(B \cup C = \\{x, z\\}\)

Step 2: Compute the Cartesian Product A x (B ∪ C)

Now that we have the union of sets B and C, we need to compute the Cartesian product of set A and the union set. The Cartesian product of two sets A and B is a set of ordered pairs (a, b) such that a is in A and b is in B. Set A = {b, c}, and the union set is B ∪ C = {x, z}. Thus, the Cartesian product is: \(A \times (B \cup C) = A \times \\{x, z\\}\)

Step 3: List the Ordered Pairs

To find the Cartesian product, we need to create ordered pairs with one element from set A and one element from the union set (B ∪ C). Let's list all the possible ordered pairs. \(A \times (B \cup C) = \\{(b,x), (b,z), (c,x), (c,z)\\}\)

Step 4: Write the Final Answer

Finally, we have calculated the Cartesian product of set A and the union of sets B and C, and it is: \(A \times (B \cup C) = \\{(b,x), (b,z), (c,x), (c,z)\\}\)

Key Concepts

Cartesian Product

The Cartesian product is a fundamental concept in set theory, which involves two sets. It's a way of pairing every element of one set with every element of another set. To understand this better, imagine you have two sets, A and B.
The Cartesian product, denoted as \(A \times B\), is the set of all possible ordered pairs where the first element is from A, and the second element is from B.

Here's how it works:

• Take each element from the first set, A.
• Pair it with every element from the second set, B.
• Each pair is called an "ordered pair."

This process results in a new set containing all these ordered pairs.
For instance, if \(A = \{1, 2\}\) and \(B = \{x, y\}\), the Cartesian product \(A \times B\) will result in the set \{(1, x), (1, y), (2, x), (2, y)\}.
Cartesian products are not just limited to numbers and letters—they can be used with any sets.

Union of Sets

The union of sets is a simple yet powerful concept that combines all the elements of two or more sets. If you have two sets, B and C, the union of these sets, denoted as \(B \cup C\), is the set consisting of all elements that are in either B or C, or in both.

• The result contains each element only once, even if it appears in both sets.
• This operation considers all unique elements from the involved sets.

For example, suppose \(B = \{x\}\) and \(C = \{x, z\}\). Then the union \(B \cup C\) equals \{x, z\}, because "x" is in both sets and "z" is in C.
Understanding the union is useful in many areas of mathematics and computer science, especially when dealing with elements spread across multiple groups.

Ordered Pairs

An ordered pair is a fundamental building block in the definition of Cartesian products. It is a pair of elements arranged in a specific sequence, typically written as \((a, b)\), where "a" is the first element and "b" is the second.
Ordered pairs are crucial because they allow you to distinguish the first element from the second, maintaining the sequence of pairing. This differs from a set, where the order of elements does not matter.
For example, \((b, x)\) is considered entirely different from \((x, b)\) because the position of "b" and "x" are switched.

• Each pair combines elements from different sets.
• It preserves the specific order in which elements are combined.

In the context of a Cartesian product, ordered pairs connect elements from each of the sets involved, highlighting their relational correspondence.