Let's solve the problem step by step. We're given:

• Universal set $U = \{a, bcde, f, g, h, i\}$
• Set $A = \{a, b, f, h, i\}$
• Set $B = \{a, b, c, d, f, h\}$
• Set $C = \{b, c, d, e, g, h\}$

We need to find the elements in the set $(A \cap C) \cup (B \cap A)$.

Step 1: Find $A \cap C$

The intersection of sets $A$ and $C$ (i.e., elements common to both sets): $A \cap C = \{b, h\}$

Step 2: Find $B \cap A$

The intersection of sets $B$ and $A$ (i.e., elements common to both sets): $B \cap A = \{a, b, f, h\}$

Step 3: Find $(A \cap C) \cup (B \cap A)$

Now, take the union of the results from the previous steps: $(A \cap C) \cup (B \cap A) = \{b, h\} \cup \{a, b, f, h\} = \{a, b, f, h\}$

Final Answer:

The elements in the set $(A \cap C) \cup (B \cap A)$ are: $\{a, b, f, h\}$

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Would you like more details or have any questions?

Here are 5 related questions to expand on this:

1. How would the result change if sets $A$, $B$, or $C$ were modified?
2. Can you explain how the union and intersection operations work on sets?
3. What happens when we take the intersection of more than two sets?
4. How do you find the complement of a set relative to the universal set?
5. How would the solution change if we were asked to find $(A \cup B) \cap C$?

Tip: When working with sets, Venn diagrams can be very useful for visualizing operations like union and intersection!