Problem 38 - Algebra and Trigonometry

Question

\(35-38=\) Find the indicated set if \(A=\\{1,2,3,4,5,6,7\\} \quad B=\\{2,4,6,8\\}\) \(C=\\{7,8,9,10\\}\) $$ \begin{array}{ll}{\text { (a) } A \cup B \cup C} & {\text { (b) } A \cap B \cap C}\end{array} $$

Step-by-Step Solution

Answer

(a) \( A \cup B \cup C = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \); (b) \( A \cap B \cap C = \emptyset \).

Step 1: Find Union of A, B, and C

The union of sets is a set containing all elements from each of the sets without repeating any element. To find \( A \cup B \cup C \), we combine all the elements from sets \( A \), \( B \), and \( C \): - Set \( A = \{1, 2, 3, 4, 5, 6, 7\} \) - Set \( B = \{2, 4, 6, 8\} \) - Set \( C = \{7, 8, 9, 10\} \) Combine all elements: \( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \). So, \( A \cup B \cup C = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \).

Step 2: Find Intersection of A, B, and C

The intersection of sets is a set containing only the elements that are present in all the sets. To find \( A \cap B \cap C \), look for elements common to sets \( A \), \( B \), and \( C \): - Set \( A = \{1, 2, 3, 4, 5, 6, 7\} \) - Set \( B = \{2, 4, 6, 8\} \) - Set \( C = \{7, 8, 9, 10\} \) Since there are no elements that are present in all three sets, the result is an empty set, denoted as \( \emptyset \). Thus, \( A \cap B \cap C = \emptyset \).

Key Concepts

Union of Sets

Understanding the union of sets is a fundamental concept in set theory. When you think of the union of sets, imagine gathering all the items from each set and putting them together into one bigger set. You don't need to worry about duplicates; each element is included only once.

For example, if we take the sets from our problem:

• Set \( A = \{1, 2, 3, 4, 5, 6, 7\} \)
• Set \( B = \{2, 4, 6, 8\} \)
• Set \( C = \{7, 8, 9, 10\} \)

To find \( A \cup B \cup C \), you combine all the unique elements from these sets:

• From \( A \): 1, 2, 3, 4, 5, 6, 7
• From \( B \): 2, 4, 6, 8
• From \( C \): 7, 8, 9, 10

Putting them all together gives us \( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \).

Remember, the union is all about combination and inclusion of every element even if it appears in multiple sets, without repetition. This simple operation allows you to see all the possible elements from the sets in one holistic view.

Intersection of Sets

The intersection of sets builds on a different principle than unions. Here, you're searching for common ground—elements that appear in every set you're comparing.

Consider our sets again:

• Set \( A = \{1, 2, 3, 4, 5, 6, 7\} \)
• Set \( B = \{2, 4, 6, 8\} \)
• Set \( C = \{7, 8, 9, 10\} \)

To compute \( A \cap B \cap C \), we look for elements that are present in \( A \), \( B \), and \( C \). Upon examining these sets, you'll notice that there is no single number that all three sets share.

Therefore, the intersection is empty, meaning these sets have no shared elements when considered together. This results in what we call an empty set, denoted as \( \emptyset \). Intersections are great for finding overlap but sometimes, like here, you find the sets just don't overlap at all.

Empty Set

The empty set is a fascinating aspect of set theory and can be initially counterintuitive. It's simply a set that contains no elements at all. In mathematical terms, it is expressed as \( \emptyset \) or sometimes just \( \{ \} \).

In the example of intersection of \( A \), \( B \), and \( C \), we found \( A \cap B \cap C \) to be \( \emptyset \) because no single element was present in all three sets. This is the classic example of an empty set—where nothing overlaps.

The empty set is unique because it serves as a placeholder in numerous mathematical proofs and processes. Understanding that a set can indeed "exist" without containing any elements helps solidify your comprehension of basic mathematical principles. Use the concept of an empty set to recognize when no commonality is present across the sets you're working with.