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.