If U= {a,e,I,o,u),A={a,e,i} , B = {e,o,u},
C = {a,I,u}, then :
(i) What is `AuuU`?
(iii) What is `AcapU`?
(iii) What is `Auuphi`?
(iv) What is `Acapphi`?

Answer

To solve the given question step by step, let's analyze each part:

### Given Sets:
- Universal Set \( U = \{ a, e, i, o, u \} \)
- Set \( A = \{ a, e, i \} \)
- Set \( B = \{ e, o, u \} \)
- Set \( C = \{ a, i, u \} \)

### (i) What is \( A \cup U \)?

**Solution:**
To find the union of set \( A \) with the universal set \( U \), we combine all unique elements from both sets.

1. Elements of \( A \): \( \{ a, e, i \} \)
2. Elements of \( U \): \( \{ a, e, i, o, u \} \)

Now, combine these elements:
- \( A \cup U = \{ a, e, i, o, u \} \)

**Final Answer:** \( A \cup U = \{ a, e, i, o, u \} \)

### (ii) What is \( A \cap U \)?

**Solution:**
To find the intersection of set \( A \) with the universal set \( U \), we identify the common elements in both sets.

1. Elements of \( A \): \( \{ a, e, i \} \)
2. Elements of \( U \): \( \{ a, e, i, o, u \} \)

Now, find the common elements:
- \( A \cap U = \{ a, e, i \} \)

**Final Answer:** \( A \cap U = \{ a, e, i \} \)

### (iii) What is \( A \cup \phi \)?

**Solution:**
To find the union of set \( A \) with the empty set \( \phi \), we include all elements of \( A \) since the empty set has no elements.

1. Elements of \( A \): \( \{ a, e, i \} \)
2. Elements of \( \phi \): \( \{ \} \)

Now, combine these elements:
- \( A \cup \phi = \{ a, e, i \} \)

**Final Answer:** \( A \cup \phi = \{ a, e, i \} \)

### (iv) What is \( A \cap \phi \)?

**Solution:**
To find the intersection of set \( A \) with the empty set \( \phi \), we look for common elements. Since the empty set has no elements, there will be no common elements.

1. Elements of \( A \): \( \{ a, e, i \} \)
2. Elements of \( \phi \): \( \{ \} \)

Thus:
- \( A \cap \phi = \{ \} \) (or \( \phi \))

**Final Answer:** \( A \cap \phi = \phi \)

### Summary of Answers:
1. \( A \cup U = \{ a, e, i, o, u \} \)
2. \( A \cap U = \{ a, e, i \} \)
3. \( A \cup \phi = \{ a, e, i \} \)
4. \( A \cap \phi = \phi \)