We are given the following sets:

• $U = \{ 3, 7, 8, 12, 11, 17, 20, 22 \}$ (the universal set),
• $A = \{ 3, 20, 22 \}$,
• $C = \{ 7, 8, 12, 11, 17 \}$.

The task is to find $A \cup C'$, where $C'$ denotes the complement of set $C$ relative to the universal set $U$.

Steps to Solve:

1. Find $C'$ (the complement of $C$): The complement of set $C$ consists of all the elements in $U$ that are not in $C$. So, $C' = U - C = \{ 3, 7, 8, 12, 11, 17, 20, 22 \} - \{ 7, 8, 12, 11, 17 \} = \{ 3, 20, 22 \}.$

2. Find $A \cup C'$ (the union of $A$ and $C'$): The union of two sets contains all the elements from both sets without repetition. We have: $A = \{ 3, 20, 22 \}, \quad C' = \{ 3, 20, 22 \}.$ Therefore, the union is: $A \cup C' = \{ 3, 20, 22 \} \cup \{ 3, 20, 22 \} = \{ 3, 20, 22 \}.$

Final Answer:

$A \cup C' = \{ 3, 20, 22 \}.$

If you need further clarification or additional examples, feel free to ask!

Additional Questions:

1. What does the complement of a set mean in set theory?
2. How do you calculate the complement of a set relative to another set?
3. What is the difference between union and intersection in set theory?
4. How would the answer change if the universal set $U$ were different?
5. Can you give an example where the union of two sets is empty?

Tip:

When working with set operations, always carefully define the universal set, as the complement depends on it!