If `A = {1, 4, 7, 8},B = {4, 6, 8, 9} andC = {3, 4, 5, 7}` be three subsets of a universal set ` U = { 1, 2, 3, 4, 5 , 6 , 7 , 8 , 9 }`. Then, ` A cup ( B cap C ' )` is equal to

Options

A. `{1, 6, 7, 8, 9}`
B. `{1, 6, 7, 8, 9, 3}`
C. `{1, 4, 6, 7, 8, 9}`
D. None of these

Answer

To solve the problem step by step, we will follow the operations defined in the question.

### Step 1: Identify the universal set and subsets
We have the following sets:
- Universal set \( U = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \} \)
- Set \( A = \{ 1, 4, 7, 8 \} \)
- Set \( B = \{ 4, 6, 8, 9 \} \)
- Set \( C = \{ 3, 4, 5, 7 \} \)

### Step 2: Find the complement of set \( C \) (denoted as \( C' \))
The complement of set \( C \) is found by subtracting the elements of \( C \) from the universal set \( U \).

\[ C' = U - C = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \} - \{ 3, 4, 5, 7 \} \]
This results in:
\[ C' = \{ 1, 2, 6, 8, 9 \} \]

### Step 3: Find the intersection of sets \( B \) and \( C' \) (denoted as \( B \cap C' \))
Now we will find the common elements between sets \( B \) and \( C' \).

\[ B = \{ 4, 6, 8, 9 \} \]
\[ C' = \{ 1, 2, 6, 8, 9 \} \]
The common elements are:
\[ B \cap C' = \{ 6, 8, 9 \} \]

### Step 4: Find the union of set \( A \) and the result from step 3 (denoted as \( A \cup (B \cap C') \))
Now we will find the union of set \( A \) and the intersection result from step 3.

\[ A = \{ 1, 4, 7, 8 \} \]
\[ B \cap C' = \{ 6, 8, 9 \} \]
The union is:
\[ A \cup (B \cap C') = \{ 1, 4, 7, 8 \} \cup \{ 6, 8, 9 \} \]
Combining these sets, we have:
\[ A \cup (B \cap C') = \{ 1, 4, 6, 7, 8, 9 \} \]

### Final Result
Thus, the final result is:
\[ A \cup (B \cap C') = \{ 1, 4, 6, 7, 8, 9 \} \]

---