Given the sets:

• $A = \{ r, e, a, s, o, n, i, g \}$
• $B = \{ m, a, t, h, e, t, i, c, l \}$
• $C =$ the set of vowels, i.e., $C = \{ a, e, i, o, u \}$

Let’s calculate each of the expressions:

1. $A \cup B \cup C$

The union of sets $A$, $B$, and $C$ includes all elements from each set, without duplicates. So,

$A \cup B \cup C = \{ r, e, a, s, o, n, i, g, m, t, h, c, l, u \}$

Thus, $A \cup B \cup C = \{ r, e, a, s, o, n, i, g, m, t, h, c, l, u \}$.

2. $A \cap B$

The intersection of $A$ and $B$ includes only elements that are in both sets.

$A \cap B = \{ e, a, i \}$

So, $A \cap B = \{ e, a, i \}$.

3. $C^c$

$C^c$ is the complement of $C$, meaning it includes all elements in the universal set that are not in $C$. Assuming the universal set includes only letters in $A \cup B$, we get:

$C^c = \{ r, s, n, g, m, t, h, c, l \}$

Thus, $C^c = \{ r, s, n, g, m, t, h, c, l \}$.

---

Would you like further explanation on any of these steps?

Here are some related questions to explore:

1. How would the union of two sets change if one set were empty?
2. What happens to the intersection if two sets are disjoint?
3. How does the complement of a set depend on the universal set?
4. What if we add more vowels to set $C$ – how would $A \cup B \cup C$ change?
5. Can the union or intersection be associative in set theory, and if so, how?

Tip: When calculating the union or intersection of sets, remember to include each unique element only once, and always verify if any elements overlap for intersections.