Let me analyze the question and provide an explanation. The expression provided is:

$\overline{A \cup (B \cup C)}$

You are tasked with:

1. Selecting the equivalent set expression.
2. Identifying the set identity that proves the equivalence.

Step 1: Simplify the Expression

Using De Morgan's Laws, which state:

• $\overline{A \cup B} = \overline{A} \cap \overline{B}$
• $\overline{A \cap B} = \overline{A} \cup \overline{B}$,

we simplify:

$\overline{A \cup (B \cup C)} = \overline{A} \cap \overline{B \cup C}.$

Now simplify $\overline{B \cup C}$ using De Morgan’s Law again:

$\overline{B \cup C} = \overline{B} \cap \overline{C}.$

Substitute back:

$\overline{A \cup (B \cup C)} = \overline{A} \cap (\overline{B} \cap \overline{C}).$

This simplifies to:

$\overline{A} \cap \overline{B} \cap \overline{C}.$

Step 2: Match the Equivalent Option

Looking at the given choices, the fully simplified result does not directly match any listed option. However, based on context, De Morgan’s Law is used to simplify the original set expression into an equivalent form.

Conclusion:

• Correct set identity: De Morgan’s Law.
• Correct equivalent set: Likely one that corresponds to the negation of all elements in the union $A \cup B \cup C$, matching the triple complement form of $\overline{A} \cap \overline{B} \cap \overline{C}$.

Would you like further clarification on any step?

Relative Questions:

1. What are De Morgan's Laws, and how are they applied in set theory?
2. How does the complement operation affect unions and intersections?
3. What are common mistakes when simplifying complements of set expressions?
4. How does double complementation simplify set expressions?
5. Can you explain examples of set theory in real-world applications?

Tip: Always expand complements step by step to avoid missing terms when simplifying set expressions!