The question asks which region(s) is/are shaded in $A' \cup C'$.

To solve this, let’s break it down:

1. Set Definitions:

   • $A'$: This is the complement of set $A$, meaning all regions outside of $A$.
   • $C'$: This is the complement of set $C$, meaning all regions outside of $C$.
   • $A' \cup C'$: This is the union of $A'$ and $C'$, meaning any region that is either outside of $A$ or outside of $C$.

2. Region Breakdown:

   • Region $I$: Inside only $A$.
   • Region $II$: Inside both $A$ and $B$.
   • Region $III$: Inside only $B$.
   • Region $IV$: Inside both $A$ and $C$.
   • Region $V$: Inside $A$, $B$, and $C$ (the intersection).
   • Region $VI$: Inside both $B$ and $C$.
   • Region $VII$: Inside only $C$.
   • Region $VIII$: Outside all sets (neither $A$, $B$, nor $C$).

3. Regions in $A' \cup C'$:

   • $A'$ includes regions: $III, VI, VII, VIII$.
   • $C'$ includes regions: $I, II, III, VIII$.
   • The union $A' \cup C'$ includes any region that is in either $A'$ or $C'$. Therefore, the regions are $I, II, III, VI, VII, VIII$.

Answer:

The regions that are shaded in $A' \cup C'$ are I, II, III, VI, VII, VIII.

Would you like more details on this? Here are 5 related questions:

1. What is the difference between the union and intersection of two sets?
2. How do you determine the complement of a set in a Venn diagram?
3. Can you explain the set operations involved in the question?
4. What are the practical applications of Venn diagrams?
5. How would the answer change if the operation was $A \cup C$ instead of $A' \cup C'$?

Tip: Always remember that the complement of a set includes all the elements not in that set.