https://file.aiquickdraw.com/mathbot/file/17259408122209fuxnfcx.jpg

We are tasked with finding the set \((A \cup B)' \cap C\), given the following sets:

- \( U = \{1, 2, 3, \dots, 8\} \) (the universal set)
- \( A = \{1, 2, 5, 7\} \)
- \( B = \{1, 2, 3\} \)
- \( C = \{1, 2, 3, 4, 5\} \)

Let's break it down:

### Step 1: Calculate \( A \cup B \)
The union of sets \(A\) and \(B\) includes all elements that are in either \(A\) or \(B\). So:
\[
A \cup B = \{1, 2, 3, 5, 7\}
\]

### Step 2: Calculate \( (A \cup B)' \)
The complement of \( A \cup B \) relative to the universal set \( U \) includes all elements that are in \( U \) but not in \( A \cup B \). So:
\[
(A \cup B)' = U - (A \cup B) = \{4, 6, 8\}
\]

### Step 3: Calculate \( (A \cup B)' \cap C \)
Now, we find the intersection of \( (A \cup B)' \) and \( C \), which includes elements that are in both sets:
\[
(A \cup B)' = \{4, 6, 8\}
\]
\[
C = \{1, 2, 3, 4, 5\}
\]
The common element between these two sets is \( \{4\} \).

### Final Answer:
\[
(A \cup B)' \cap C = \{4\}
\]

Thus, the correct choice is **A**: \((A \cup B)' \cap C = \{4\}\).

Do you have any questions or would like further details on this?

Here are 5 related questions to deepen your understanding:

1. How would the result change if the universal set \(U\) were different?
2. What is the difference between union and intersection of sets?
3. How does the complement of a set affect the elements being considered?
4. What is the result of \( A' \cap B \) for these sets?
5. How would you visualize these operations using a Venn diagram?

**Tip**: The complement of a set relative to a universal set removes all elements from the universal set that are included in the original set.

Given the sets U = {1, 2, 3, ..., 8}, A = {1, 2, 5, 7}, B = {1, 2, 3}, C = {1, 2, 3, 4, 5}, find the set (A ∪ B)' ∩ C.

Solve the set operation problem (A ∪ B)' ∩ C using the sets U = {1, 2, 3, ..., 8}, A = {1, 2, 5, 7}, B = {1, 2, 3}, and C = {1, 2, 3, 4, 5}. Learn about union, intersection, and complement in set theory. This problem is suitable for Grades 8-10 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation