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Let's break down the problem step by step:

We are given the following sets:

- \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \) (Universal set)
- \( A = \{1, 3, 5, 7, 9\} \)
- \( B = \{2, 4, 5, 6, 8\} \)
- \( C = \{3, 6, 7, 9\} \)

The problem asks for \( A' \cup (B \cap C') \), where:
- \( A' \) is the complement of set \( A \) (elements in \( U \) but not in \( A \)),
- \( C' \) is the complement of set \( C \),
- \( B \cap C' \) is the intersection of sets \( B \) and \( C' \),
- Then, we take the union \( A' \cup (B \cap C') \).

### Step 1: Find \( A' \)
The complement of \( A \) with respect to \( U \) is:
\[
A' = U - A = \{2, 4, 6, 8, 10\}
\]

### Step 2: Find \( C' \)
The complement of \( C \) with respect to \( U \) is:
\[
C' = U - C = \{1, 2, 4, 5, 8, 10\}
\]

### Step 3: Find \( B \cap C' \)
Now, find the intersection of \( B \) and \( C' \):
\[
B = \{2, 4, 5, 6, 8\}
\]
\[
C' = \{1, 2, 4, 5, 8, 10\}
\]
The intersection \( B \cap C' \) is:
\[
B \cap C' = \{2, 4, 5, 8\}
\]

### Step 4: Find \( A' \cup (B \cap C') \)
Now, take the union of \( A' \) and \( B \cap C' \):
\[
A' = \{2, 4, 6, 8, 10\}
\]
\[
B \cap C' = \{2, 4, 5, 8\}
\]
The union \( A' \cup (B \cap C') \) is:
\[
A' \cup (B \cap C') = \{2, 4, 5, 6, 8, 10\}
\]

Thus, the result of \( A' \cup (B \cap C') \) is \( \{2, 4, 5, 6, 8, 10\} \).

Would you like a deeper explanation of any step? Here are some related questions you can explore:

1. What is the difference between intersection and union of sets?
2. How do you find the complement of a set in a universal set?
3. Can a union of two sets be smaller than their intersection?
4. What is the role of the universal set in set operations?
5. How does De Morgan's Law apply to set complements?

**Tip:** Always make sure to differentiate between the union and intersection of sets when solving such problems.

Find A' ∪ (B ∩ C'), where U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 3, 5, 7, 9}, B = {2, 4, 5, 6, 8}, and C = {3, 6, 7, 9}.

Solve the set theory problem A' ∪ (B ∩ C') where A = {1, 3, 5, 7, 9}, B = {2, 4, 5, 6, 8}, and C = {3, 6, 7, 9}. This step-by-step solution covers key concepts like complements, intersections, and unions. Suitable for Grades 8-10.

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