If U = {a,b,c,d,e,f,g,h} , find the complement of the following sets:
A= {a,b,c} , B ={d,e,f,g} , C ={a,c,eg}, D = {f,g,h,a}.

Answer

To find the complement of the sets A, B, C, and D with respect to the universal set U, we will follow these steps:

### Given:
- 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 \( D = \{f, g, h, a\} \)

### Step 1: Find the complement of set A
The complement of set A, denoted as \( A' \), is found by removing the elements of A from U.

\[ A' = U - A = \{a, b, c, d, e, f, g, h\} - \{a, b, c\} \]
\[ A' = \{d, e, f, g, h\} \]

### Step 2: Find the complement of set B
The complement of set B, denoted as \( B' \), is found by removing the elements of B from U.

\[ B' = U - B = \{a, b, c, d, e, f, g, h\} - \{d, e, f, g\} \]
\[ B' = \{a, b, c, h\} \]

### Step 3: Find the complement of set C
The complement of set C, denoted as \( C' \), is found by removing the elements of C from U.

\[ C' = U - C = \{a, b, c, d, e, f, g, h\} - \{a, c, e, g\} \]
\[ C' = \{b, d, f, h\} \]

### Step 4: Find the complement of set D
The complement of set D, denoted as \( D' \), is found by removing the elements of D from U.

\[ D' = U - D = \{a, b, c, d, e, f, g, h\} - \{f, g, h, a\} \]
\[ D' = \{b, c, d, e\} \]

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

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