Problem 17 - Elementary and Intermediate Algebra

Question

Let \(A=\\{0,1,2,3,4,5,6\\}, B=\\{4,6,8,10\\}\) \(C=\\{-3,-1,0,1,2\\},\) and \(D=\\{-3,1,2,5,8\\}\) Find each set. $$ A \cap B $$

Step-by-Step Solution

Answer

The intersection of sets A and B is \( \{4, 6\} \).

Step 1: Identify Elements in Sets A and B

First, list the elements of set A: \(A = \{0, 1, 2, 3, 4, 5, 6\}\) and set B: \(B = \{4, 6, 8, 10\}\).

Step 2: Find Common Elements

Compare the two sets, \(A\) and \(B\), and find the elements that appear in both. These elements will form the intersection of the two sets.

Step 3: Form the Intersection Set

The elements \(4\) and \(6\) are present in both sets \(A\) and \(B\). Therefore, the intersection of sets \(A\) and \(B\) is \(A \cap B = \{4, 6\}\).

Key Concepts

Intersection of Sets

The intersection of sets is a fundamental concept in set theory. It refers to the operation of finding common elements that exist in two or more sets. If you imagine two circles that partially overlap, the overlapping area represents the intersection of those sets. This overlap includes all elements that the sets share. To denote the intersection of two sets, we use the symbol \( \cap \). For example, if you have sets \( A \) and \( B \), their intersection is written as \( A \cap B \).

Consider understanding this through a straightforward process:

• First, list all elements of each set.
• Next, identify elements that appear in both sets.
• Finally, those common elements form the new set, which is the intersection.

In the example provided, the intersection \( A \cap B \) consisted of finding common numbers in sets \( A = \{0, 1, 2, 3, 4, 5, 6\} \) and \( B = \{4, 6, 8, 10\} \). The result is \( A \cap B = \{4, 6\} \), where both 4 and 6 appear in each set.

Common Elements in Sets

Identifying common elements between two sets is crucial for understanding their intersection. Common elements are simply those items that appear in both of the compared sets. These elements help you determine how the sets are related.

Here's a simple guide for finding common elements:

• Write down each element of both sets separately.
• Scan through the lists to see which elements occur in both lists.
• Record these matches as your common elements.

In our exercise, the task was to identify common elements between \( A = \{0, 1, 2, 3, 4, 5, 6\} \) and \( B = \{4, 6, 8, 10\} \). Comparing these two, we found 4 and 6 appear in both lists. So, these are the common elements which make up the set \( A \cap B \).

Element Comparison

Element comparison is an essential step in determining both the intersection of sets and identifying common elements. You compare elements of one set with another to check if they exist in both sets. It's like a matching game where you search for similarities between two lists.

To successfully compare elements:

• Write each element from each set on separate lists.
• Look at each element of the first set and see if it's present in the second set.
• If it is, note it down as part of your intersection.

In practice, this was done by comparing elements of sets \( A = \{0, 1, 2, 3, 4, 5, 6\} \) and \( B = \{4, 6, 8, 10\} \). By checking each element of \( A \) against \( B \), we quickly identified that 4 and 6 were present in both, forming the intersection \( A \cap B = \{4, 6\} \). Through this comparison, clarity is brought to the relationship between sets and their shared attributes.