Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {0,1, 2, 3}, B = {3, 4, 7, 8, 9}, and
C = {3, 4, 5, 8, 9}. Find Ac n (BUC).

We are given the following sets:

- Universal set \( U = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \)
- \( A = \{0, 1, 2, 3\} \)
- \( B = \{3, 4, 7, 8, 9\} \)
- \( C = \{3, 4, 5, 8, 9\} \)

We are tasked with finding \( A^c \cap (B \cup C) \).

### Step 1: Find the complement of \( A \), denoted \( A^c \)
The complement of \( A \) refers to all elements in the universal set \( U \) that are not in \( A \). Thus,

\[
A^c = U - A = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} - \{0, 1, 2, 3\} = \{4, 5, 6, 7, 8, 9\}
\]

### Step 2: Find \( B \cup C \)
The union \( B \cup C \) consists of all elements that are in \( B \), in \( C \), or in both. Thus,

\[
B \cup C = \{3, 4, 7, 8, 9\} \cup \{3, 4, 5, 8, 9\} = \{3, 4, 5, 7, 8, 9\}
\]

### Step 3: Find \( A^c \cap (B \cup C) \)
The intersection \( A^c \cap (B \cup C) \) consists of all elements that are in both \( A^c \) and \( B \cup C \). Thus,

\[
A^c = \{4, 5, 6, 7, 8, 9\}
\]
\[
B \cup C = \{3, 4, 5, 7, 8, 9\}
\]

The intersection of these two sets is:

\[
A^c \cap (B \cup C) = \{4, 5, 7, 8, 9\}
\]

### Final Answer:
\[
A^c \cap (B \cup C) = \{4, 5, 7, 8, 9\}
\]

Would you like a detailed breakdown of any part? Here are five related questions:

1. How would the result change if set \( A \) had different elements?
2. What is the union of \( A \) and \( B \)?
3. What is the complement of \( B \) relative to \( U \)?
4. Can you explain the properties of set complements and intersections?
5. How can Venn diagrams help visualize this problem?

**Tip:** The complement of a set always relates to the universal set it belongs to; be sure to check the definition of the universal set carefully.

Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {0,1, 2, 3}, B = {3, 4, 7, 8, 9}, and C = {3, 4, 5, 8, 9}. Find Ac n (B U C).

Solve a set theory problem involving the complement of A and the union of B and C. Learn how to compute Ac ∩ (B ∪ C) step-by-step using key concepts like set complements, union, and intersection. Suitable for students in Grades 9-10.

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