https://file.aiquickdraw.com/mathbot/file/1725548805876mi4nf8ay.jpg

The image you uploaded contains a Venn diagram with two sets \( A \) and \( B \), labeled in a figure. The task is to draw a similar diagram and shade the regions representing the following sets:

1. \( A' \)
2. \( (A \cup B)' \)
3. \( (A' \cup B')' \)

### Explanation:

1. **\( A' \)**: This represents the complement of set \( A \), meaning all elements that are **not** in set \( A \). In the Venn diagram, this would be the entire region outside of circle \( A \) (but inside the universal set).

2. **\( (A \cup B)' \)**: This represents the complement of the union of sets \( A \) and \( B \). The union \( A \cup B \) is the area covered by both \( A \) and \( B \). The complement \( (A \cup B)' \) is everything outside of both sets, i.e., all areas not covered by \( A \cup B \).

3. **\( (A' \cup B')' \)**: This can be simplified using **De Morgan's law**:
   \[
   (A' \cup B')' = A \cap B
   \]
   This means the intersection of sets \( A \) and \( B \), i.e., the region common to both sets \( A \) and \( B \). You would shade the overlapping part of the two circles.

If you'd like, I can guide you on how to draw the Venn diagrams for each case.

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Would you like more detailed explanations for these concepts or solutions? 

Here are 5 related questions to deepen your understanding:
1. What is the complement of a set, and how is it represented in a Venn diagram?
2. How do you use De Morgan's laws to simplify expressions involving set complements?
3. What is the difference between a union and an intersection in set theory?
4. How do you visually distinguish between \( A \cup B \) and \( A \cap B \) in a Venn diagram?
5. How can Venn diagrams be used to solve more complex set theory problems?

**Tip**: De Morgan's laws are essential in set theory to simplify expressions involving unions and intersections, especially when working with complements.

Draw a Venn diagram and shade the regions representing the following sets: (1) A', (2) (A ∪ B)', (3) (A' ∪ B')'.

This page provides a detailed explanation of how to draw Venn diagrams representing the complement of sets, unions, and intersections, including the use of De Morgan's laws. Suitable for students in Grades 9-12 studying set theory.

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