https://file.aiquickdraw.com/mathbot/file/1727862345732kxd8k38n.jpg

Let's go through the image step by step and solve the problems.

### Part A: Venn Diagram

#### 1. \( U \) (Universal Set)
The universal set \( U \) consists of all elements inside and outside the circles, which in this case are the letters:
\[
U = \{a, b, c, d, e, f, g, h, i, j\}
\]

#### 2. \( A' \cup B' \) 
- \( A' \) is the complement of set \( A \), meaning all elements not in \( A \). The elements outside \( A \) are \( \{d, e, f, g, h, i, j\} \).
- \( B' \) is the complement of set \( B \), meaning all elements not in \( B \). The elements outside \( B \) are \( \{a, b, h, i, j\} \).

The union \( A' \cup B' \) consists of all elements in \( A' \) or \( B' \):
\[
A' \cup B' = \{a, b, d, e, f, g, h, i, j\}
\]

#### 3. \( A' \cap B' \) 
The intersection \( A' \cap B' \) consists of elements that are both in \( A' \) and \( B' \). This would be the elements outside both \( A \) and \( B \), which are:
\[
A' \cap B' = \{h, i, j\}
\]

#### 4. \( A' \cap B \) 
This is the intersection of the complement of \( A \) and set \( B \). So, we are looking for elements in \( B \) that are not in \( A \):
\[
A' \cap B = \{d, f\}
\]

#### 5. \( A \cup B' \)
This is the union of set \( A \) and the complement of \( B \). The elements in \( A \) are \( \{a, b, c\} \) and the elements in \( B' \) are \( \{a, b, h, i, j\} \). The union is:
\[
A \cup B' = \{a, b, c, h, i, j\}
\]

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### Part B: Given sets \( A = \{2, 4, 5, 8\} \), \( B = \{3, 4, 6, 8\} \), and \( U = \{1, 2, 3, 4, 5, 6, 7, 8\} \)

#### 1. \( A' \) 
\( A' \) is the complement of set \( A \), meaning all elements in the universal set \( U \) that are not in \( A \). So:
\[
A' = \{1, 3, 6, 7\}
\]

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Would you like further clarification on any of these solutions? Let me know if you'd like more details or if you have any other questions.

Here are 5 related questions you can consider:
1. What is the intersection of \( A \cap B \) from Part B?
2. What is \( B' \cup A' \) using the sets from Part B?
3. Can you visualize how the complement of a set affects its union or intersection?
4. How would you solve for \( A \cap B' \) in Part B?
5. How can you verify your Venn diagram answers by comparing them with the actual sets?

**Tip:** In Venn diagrams, intersections represent the common elements between sets, while unions gather elements from both sets.

List the elements of the following sets from the figure shown: U, A'∪B', A'∩B', A'∩B, A∪B', and reflect the sets A = {2, 4, 5, 8}, B = {3, 4, 6, 8}, U = {1, 2, 3, 4, 5, 6, 7, 8}.

This problem covers solving set operations like A'∪B', A'∩B', and A∪B' using Venn diagrams and set theory. It includes detailed steps on complement, union, and intersection, suitable for Grades 9-11 students.

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