Problem Analysis

The question involves basic set operations with the following details:

• Universal set $U = \{5, 6, 8, 1, p, q, r\}$
• Set $D$: $D = \{6, 8, p, q\}$
• Two tasks:
  1. Find $D \cup U$ (the union of $D$ and $U$).
  2. Find $D \cap \emptyset$ (the intersection of $D$ with the empty set).

Solution:

(a) $D \cup U$:

The union of two sets combines all unique elements from both sets.

• $D = \{6, 8, p, q\}$
• $U = \{5, 6, 8, 1, p, q, r\}$

The union of $D$ and $U$ is: $D \cup U = \{5, 6, 8, 1, p, q, r\}$

(b) $D \cap \emptyset$:

The intersection of any set with the empty set $\emptyset$ is always $\emptyset$, because there are no elements in common.

So: $D \cap \emptyset = \emptyset$

Final Answers:

1. (a) $D \cup U = \{5, 6, 8, 1, p, q, r\}$
2. (b) $D \cap \emptyset = \emptyset$

Would you like me to explain the union and intersection concepts further? Here are some related questions to deepen understanding:

1. What is the difference between the union and the intersection of two sets?
2. How do we compute the intersection of two sets with overlapping elements?
3. What is the complement of a set, and how does it relate to the universal set?
4. Can the union of two sets ever equal the universal set $U$?
5. What happens when the intersection of a set is taken with itself?

Tip: Remember, the empty set $\emptyset$ contains no elements and has a size of 0, making intersections with it straightforward!