Let's solve the problem step by step based on the information from the uploaded image.

Problem Breakdown:

1. Definitions:

   • $U = \{g, h, p, q, r\}$ (Universal set)
   • $D = \{g, p, q\}$

2. Tasks:

   • (a) Find $D' \cup U$, where $D'$ is the complement of $D$ (elements in $U$ but not in $D$).
   • (b) Find $D \cap \emptyset$.

Solution:

Part (a): $D' \cup U$

• Step 1: Find $D'$, the complement of $D$: $D' = U - D = \{g, h, p, q, r\} - \{g, p, q\} = \{h, r\}$

• Step 2: Take the union of $D'$ and $U$: $D' \cup U = \{h, r\} \cup \{g, h, p, q, r\} = \{g, h, p, q, r\}$

So, $D' \cup U = U = \{g, h, p, q, r\}$.

Part (b): $D \cap \emptyset$

• The intersection of any set with the empty set ($\emptyset$) is always $\emptyset$: $D \cap \emptyset = \emptyset$

Final Answers:

• (a) $D' \cup U = \{g, h, p, q, r\}$
• (b) $D \cap \emptyset = \emptyset$

Would you like further clarification on these steps or related concepts?

5 Related Questions:

1. What is the general rule for finding the complement of a set?
2. How do intersections and unions differ when applied to disjoint sets?
3. What is the union of two identical sets?
4. Can a universal set have subsets that do not include any of its elements?
5. What happens when you take the union of a set with the empty set?

Tip:

To verify set operations, always write out the elements explicitly—it helps to avoid missing or duplicating elements.