The value of `(A cup B cup C) cap (A cap B^(C)capC^(C)) cap C^(C)` is

Options

A. `(B cap C^(c))`
B. `(A cap B^(c) cap C^(C))`
C. `(B cap C)`
D. `(A cap B cap C)`

Answer

To solve the expression \((A \cup B \cup C) \cap (A \cap B^{C} \cap C^{C}) \cap C^{C}\), we will break it down step by step.

### Step 1: Understand the components of the expression
- **Union (\(\cup\))**: Combines all elements from the sets.
- **Intersection (\(\cap\))**: Only includes elements that are present in all sets.
- **Complement (\(C^{C}\))**: Includes all elements not in the set \(C\).

### Step 2: Find \(A \cup B \cup C\)
This part of the expression represents all elements that are in at least one of the sets \(A\), \(B\), or \(C\).
- **Result**:
\[ A \cup B \cup C = \text{All elements in } A, B, \text{ or } C \]

### Step 3: Find \(A \cap B^{C} \cap C^{C}\)
- **\(B^{C}\)**: This is the complement of \(B\), meaning all elements not in \(B\).
- **\(C^{C}\)**: This is the complement of \(C\), meaning all elements not in \(C\).
- The intersection \(A \cap B^{C} \cap C^{C}\) includes elements that are in \(A\) but not in \(B\) or \(C\).
- **Result**:
\[ A \cap B^{C} \cap C^{C} = \text{Elements in } A \text{ that are not in } B \text{ or } C \]

### Step 4: Find \(C^{C}\)
This is simply all elements that are not in set \(C\).
- **Result**:
\[ C^{C} = \text{All elements not in } C \]

### Step 5: Combine the results
Now we need to evaluate the entire expression:
\[ (A \cup B \cup C) \cap (A \cap B^{C} \cap C^{C}) \cap C^{C} \]
1. **First part**: \(A \cup B \cup C\) includes all elements in \(A\), \(B\), or \(C\).
2. **Second part**: \(A \cap B^{C} \cap C^{C}\) includes elements in \(A\) that are not in \(B\) or \(C\).
3. **Third part**: \(C^{C}\) includes all elements not in \(C\).

### Step 6: Find the intersection
- The intersection of these three parts will yield elements that are in \(A\) but not in \(B\) or \(C\), and also not in \(C\).
- Therefore, the result simplifies to just the elements in \(A\) that are not in \(B\) or \(C\).

### Final Result
The value of the expression is:
\[ A \cap B^{C} \cap C^{C} \]
This means the elements that are in \(A\) but not in either \(B\) or \(C\).