We are given the universal set $U = \{a, b, c, d, e, i, o, u\}$, and the sets $A = \{a, b, c, d, e\}$, $B = \{a, e, i, o, u\}$, and $C = \{i, o, u\}$. Now, let's find each set operation.

A) $A \cup C$ (Union of A and C)

The union of two sets contains all elements that are in either of the sets. So, $A \cup C = \{a, b, c, d, e\} \cup \{i, o, u\}$ Combining all elements, we get: $A \cup C = \{a, b, c, d, e, i, o, u\}$ Thus, $A \cup C = U$.

B) $A \cap C$ (Intersection of A and C)

The intersection of two sets contains only the elements that are in both sets. So, $A \cap C = \{a, b, c, d, e\} \cap \{i, o, u\}$ Since there are no common elements between $A$ and $C$, the intersection is empty: $A \cap C = \emptyset$

C) $(A \cap B)'$ (Complement of the Intersection of A and B)

First, find $A \cap B$, which includes elements common to both sets: $A \cap B = \{a, b, c, d, e\} \cap \{a, e, i, o, u\} = \{a, e\}$ Now, the complement of $A \cap B$ (denoted $(A \cap B)'$) is the set of elements in $U$ but not in $A \cap B$: $(A \cap B)' = U \setminus \{a, e\} = \{b, c, d, i, o, u\}$

D) $A' \cup B'$ (Union of Complements of A and B)

First, find the complements of $A$ and $B$.

The complement of $A$ (denoted $A'$) is the set of elements in $U$ but not in $A$: $A' = U \setminus A = \{i, o, u\}$

The complement of $B$ (denoted $B'$) is the set of elements in $U$ but not in $B$: $B' = U \setminus B = \{b, c, d\}$

Now, the union of $A'$ and $B'$ is: $A' \cup B' = \{i, o, u\} \cup \{b, c, d\} = \{b, c, d, i, o, u\}$

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Summary of Results:

A) $A \cup C = \{a, b, c, d, e, i, o, u\}$
B) $A \cap C = \emptyset$
C) $(A \cap B)' = \{b, c, d, i, o, u\}$
D) $A' \cup B' = \{b, c, d, i, o, u\}$

Would you like further details or clarifications? Here are 5 related questions to consider:

1. What is the complement of $C$ relative to $U$?
2. How would $A \cup B$ differ from $A \cup C$?
3. What happens if $A \cap B \cap C$ is computed?
4. How do the operations change if we have sets with infinite elements?
5. Can the union of any two sets be the universal set?

Tip: The union of a set with its complement always gives the universal set!