The problem asks to select the regions corresponding to the set $(A \cap B) \cup C$, referring to the Venn diagram provided.

Let's break this down:

1. $A \cap B$: This represents the intersection of sets $A$ and $B$, which means elements that are in both $A$ and $B$. In the Venn diagram, this corresponds to region V (the overlapping area of $A$ and $B$).

2. $C$: This represents all elements in set $C$. In the diagram, this includes regions IV, V, VI, and VII (the entire area within circle $C$).

3. $(A \cap B) \cup C$: This represents the union of the two sets $A \cap B$ and $C$. In terms of regions, it includes all the regions that are either in $A \cap B$ or in $C$.

So, the regions corresponding to $(A \cap B) \cup C$ are:

• Region IV
• Region V
• Region VI
• Region VII

Would you like any further explanation or clarification?

Here are some related questions to explore:

1. What would be the regions corresponding to $A \cup B$?
2. What regions represent $A \cap C$?
3. How would you represent $B \cap C$?
4. What regions would represent $(A \cup B) \cap C$?
5. How would the regions change if you were asked for $(A \cup C) \cap B$?

Tip: When solving Venn diagram problems, always work through the intersections and unions step by step to ensure accuracy.