Problem 7 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 10, list the elements of the given set if \(A=\\{0,2,4,6,8\\}, B=\\{1,2,4,8\\}, C=\\{1,3,5,7,9\\}\), and \(D=\) \(\\{0,3,6,9\\}\). $$ B \cap D $$

Step-by-Step Solution

Answer

The intersection \( B \cap D \) is \( \emptyset \).

Step 1 - Identify the sets involved

The sets involved in the problem are given as follows: \(B=\{1,2,4,8\}\) and \(D=\{0,3,6,9\}\).

Step 2 - Understand the intersection operation

The intersection operation, denoted by \( \cap \), involves finding the common elements between two sets. In this case, we need to find the common elements between sets \(B\) and \(D\).

Step 3 - List the elements of set B

The elements of set \(B\) are: \(\{1,2,4,8\}\).

Step 4 - List the elements of set D

The elements of set \(D\) are: \(\{0,3,6,9\}\).

Step 5 - Find common elements

Compare the elements of sets \(B\) and \(D\) to identify common elements. Set \(B\) elements: \(\{1,2,4,8\}\) Set \(D\) elements: \(\{0,3,6,9\}\) It is clear that there are no common elements between sets \(B\) and \(D\).

Step 6 - Write the intersection result

Since there are no common elements between sets \(B\) and \(D\), the intersection \( B \cap D \) is the empty set \( \emptyset \).

Key Concepts

set intersection

Set intersection is a fundamental operation in set theory. It identifies the common elements shared between two sets. The symbol used for intersection is \( \cap \). When we perform a set intersection between two sets A and B, we look for all elements that are present in both sets at the same time.
For example, consider sets \( A = \{2, 4, 6\} \) and \( B = \{4, 5, 6, 7\} \). The intersection, denoted as \( A \cap B \), would include elements that are in both set A and set B. In this case, \( A \cap B = \{4, 6\}. \) When you compare the sets element by element, you see that 4 and 6 appear in both.
In our original exercise, we compared sets \( B = \{1, 2, 4, 8\} \) and \( D = \{0, 3, 6, 9\}. \) Since there were no elements common to both, their intersection is the empty set.

empty set

An empty set, denoted as \( \emptyset \) or \( \{\} \), is a set that contains no elements. It is one of the most critical and basic concepts in set theory because it serves as the identity element for the operation of union and intersection.
When the result of a set operation, like intersection, yields no shared elements, we describe the result as an empty set. In the context of our intersection example, sets \( B \) and \( D \) have no common elements, which means their intersection is the empty set \( B \cap D = \emptyset \).
It's important to note that the empty set is unique. No matter where or how it occurs, it is always the same set containing no elements.

element comparison

Element comparison is the process of examining each element from one set and comparing it with the elements in another set to find commonalities. This process is essential in operations like set intersection.
In the original exercise, to perform the set intersection \( B \cap D \), we compared each element in set \( B = \{1, 2, 4, 8\} \) with each element in set \( D = \{0, 3, 6, 9\}. \) We checked elements one by one:
\begin{itemize} \item B has 1, D has 0, 3, 6, and 9 (no match) \item B has 2, D has 0, 3, 6, and 9 (no match) \item B has 4, D has 0, 3, 6, and 9 (no match) \item B has 8, D has 0, 3, 6, and 9 (no match) \end{itemize}
Since none of the comparisons resulted in a common value, the intersection was determined to be the empty set, \( \emptyset \). Element comparison ensures that we accurately find intersections by methodically checking all possibilities.