
Let A = {3, 9}, B = {9, 19, 21}, D = {36} and S = sample space = A ∪ B ∪ D. Identify Ac ∩ B.

To solve this, we need to identify the complement of set \( A \), denoted by \( A^c \), and find its intersection with set \( B \).

### Step 1: Find the union of sets \( A \), \( B \), and \( D \).
- \( A = \{3, 9\} \)
- \( B = \{9, 19, 21\} \)
- \( D = \{36\} \)

The sample space \( S \) is the union of \( A \), \( B \), and \( D \):
\[
S = A \cup B \cup D = \{3, 9\} \cup \{9, 19, 21\} \cup \{36\} = \{3, 9, 19, 21, 36\}
\]

### Step 2: Determine the complement of \( A \), denoted by \( A^c \).
The complement \( A^c \) consists of all elements in the sample space \( S \) that are **not** in \( A \). So:
\[
A^c = S \setminus A = \{3, 9, 19, 21, 36\} \setminus \{3, 9\} = \{19, 21, 36\}
\]

### Step 3: Find the intersection \( A^c \cap B \).
Now, we calculate the intersection of \( A^c \) and \( B \):
\[
A^c = \{19, 21, 36\}, \quad B = \{9, 19, 21\}
\]
The intersection is:
\[
A^c \cap B = \{19, 21\}
\]

### Final Answer:
\[
A^c \cap B = \{19, 21\}
\]

Would you like more details or further clarifications?

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Here are 5 related questions:
1. What is the union of \( A \) and \( B \)?
2. How do you find the complement of a set?
3. What is the intersection of \( A \cap B \)?
4. If \( D = \{36, 45\} \), how does it affect \( A^c \cap B \)?
5. How would you visualize \( A^c \cap B \) using a Venn diagram?

**Tip:** To find a complement of a set, always subtract the set from the full sample space.

Let A = {3, 9}, B = {9, 19, 21}, D = {36} and S = sample space = A ∪ B ∪ D. Identify Ac ∩ B.

Learn how to find the complement of set A and its intersection with set B, given sets A, B, and D. This problem explores key set theory concepts, including union, complement, and intersection operations. Perfect for high school students in Grades 9-12.

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