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
1. **Identify the components of the expression**:
- \(A \cup B \cup C\): This represents the union of sets A, B, and C, which includes all elements in A, B, or C.
- \(A \cap B^{C} \cap C^{C}\): This represents the intersection of set A with the complements of B and C, which includes elements that are in A but not in B and not in C.
- \(C^{C}\): This represents the complement of set C, which includes all elements not in C.

### Step 2: Calculate \(A \cup B \cup C\)
2. **Calculate the union**:
- The union \(A \cup B \cup C\) includes all elements from sets A, B, and C. Therefore, it can be represented as:
\[ A \cup B \cup C = A + B + C + D + E + F + G \]
where D, E, F, and G are elements outside of A, B, and C.

### Step 3: Calculate \(A \cap B^{C} \cap C^{C}\)
3. **Calculate the intersection**:
- \(B^{C}\) is everything except B, and \(C^{C}\) is everything except C. Therefore, \(A \cap B^{C} \cap C^{C}\) will include only those elements that are in A and not in B and not in C. This can be represented as:
\[ A \cap B^{C} \cap C^{C} = A \text{ (only the part of A that is not in B or C)} \]

### Step 4: Calculate \(C^{C}\)
4. **Identify the complement of C**:
- \(C^{C}\) includes all elements that are not in C, which can be represented as:
\[ C^{C} = A + B + D + E + F + G \]

### Step 5: Combine the Results
5. **Combine the results using intersection**:
- Now, we need to find the intersection of the results from steps 2, 3, and 4:
\[ (A \cup B \cup C) \cap (A \cap B^{C} \cap C^{C}) \cap C^{C} \]
- The intersection of \(A \cup B \cup C\) with \(A \cap B^{C} \cap C^{C}\) gives us only the part of A that is not in B or C.
- Then, intersecting this with \(C^{C}\) (which includes everything except C) will still yield only the part of A that is not in C.

### Final Result
6. **Final expression**:
- Therefore, the final value of the expression is simply:
\[ A \]

### Conclusion
The value of the expression \((A \cup B \cup C) \cap (A \cap B^{C} \cap C^{C}) \cap C^{C}\) is \(A\).

---