If `U = {a , b , c , d , e , f, g, h}`, find the complements of the following sets :(i) `A = {a , b , c}` (ii) `B = {d , e , f, g]` (iii) `C = {a , c , e , g}` (iv) `D = {f, g, h , a)`

Options

A. `npi+(-1)^npi/6, n in Z`
B. `npi+(-1)^npi/3, n in Z`
C. `npipmpi/6, n in Z`
D. `npipmpi/3, n in Z`

Explanation

To find the complements of the given sets with respect to the universal set \( U = \{a, b, c, d, e, f, g, h\} \), we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Universal Set and the Sets:**
- Universal Set \( U = \{a, b, c, d, e, f, g, h\} \)
- Set \( A = \{a, b, c\} \)
- Set \( B = \{d, e, f, g\} \)
- Set \( C = \{a, c, e, g\} \)
- Set \( D = \{f, g, h, a\} \)

2. **Find the Complement of Set A:**
- The complement of set \( A \) is defined as \( U - A \).
- Elements in \( A \) are \( \{a, b, c\} \).
- Removing these from \( U \):
\[ U - A = \{d, e, f, g, h\} \]
- **Complement of A:** \( A' = \{d, e, f, g, h\} \)

3. **Find the Complement of Set B:**
- The complement of set \( B \) is defined as \( U - B \).
- Elements in \( B \) are \( \{d, e, f, g\} \).
- Removing these from \( U \):
\[ U - B = \{a, b, c, h\} \]
- **Complement of B:** \( B' = \{a, b, c, h\} \)

4. **Find the Complement of Set C:**
- The complement of set \( C \) is defined as \( U - C \).
- Elements in \( C \) are \( \{a, c, e, g\} \).
- Removing these from \( U \):
\[ U - C = \{b, d, f, h\} \]
- **Complement of C:** \( C' = \{b, d, f, h\} \)

5. **Find the Complement of Set D:**
- The complement of set \( D \) is defined as \( U - D \).
- Elements in \( D \) are \( \{f, g, h, a\} \).
- Removing these from \( U \):
\[ U - D = \{b, c, d, e\} \]
- **Complement of D:** \( D' = \{b, c, d, e\} \)

### Summary of Complements:
- \( A' = \{d, e, f, g, h\} \)
- \( B' = \{a, b, c, h\} \)
- \( C' = \{b, d, f, h\} \)
- \( D' = \{b, c, d, e\} \)