Given *U* = {a,b,c,d,e,i,o,u}, *A* = {a,b,c,d,e}, *B* = {a,e,i,o,u}, *C* = {i,o,u}, find the following sets.

A) A∪C=

B) A∩C=

C) (A∩B)′=

D) A′∪B′=

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:
\[
A' \cup B' = \{i, o, u\} \cup \{b, c, d\} = \{b, c, d, i, o, u\}
\]

---

### 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!

Given U = {a,b,c,d,e,i,o,u}, A = {a,b,c,d,e}, B = {a,e,i,o,u}, C = {i,o,u}, find the following sets.
A) A∪C=
B) A∩C=
C) (A∩B)′=
D) A′∪B′=

Solve set theory problems involving the union, intersection, and complement of sets A = {a,b,c,d,e}, B = {a,e,i,o,u}, C = {i,o,u}, with a universal set U = {a,b,c,d,e,i,o,u}. Learn how to compute A∪C, A∩C, (A∩B)', and A'∪B'. Ideal for students in Grades 9-12.

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