The set `(Auu BuuC)nn(AnnB'nnB')'nnC` is equal to

Options

A. `AnnB`
B. A
C. `BnnC'`
D. C

Answer

To solve the given set expression \( (A \cup B \cup C) \cap (A \cap B' \cap C')' \cap C' \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (A \cup B \cup C) \cap (A \cap B' \cap C')' \cap C' \]

### Step 2: Apply De Morgan's Law
Using De Morgan's Law, we can simplify \( (A \cap B' \cap C')' \):
\[ (A \cap B' \cap C')' = A' \cup (B')' \cup (C')' = A' \cup B \cup C \]

### Step 3: Substitute back into the expression
Now, substitute this back into the original expression:
\[ (A \cup B \cup C) \cap (A' \cup B \cup C) \cap C' \]

### Step 4: Distribute the intersection
We can distribute the intersection:
\[ [(A \cup B \cup C) \cap (A' \cup B \cup C)] \cap C' \]

### Step 5: Simplify the first part
Now we simplify \( (A \cup B \cup C) \cap (A' \cup B \cup C) \):
- The term \( A \) will be eliminated because \( A \cap A' = \emptyset \).
- Thus, we have:
\[ (B \cup C) \cap (A' \cup B \cup C) = B \cup C \]

### Step 6: Combine with the remaining part
Now we have:
\[ (B \cup C) \cap C' \]

### Step 7: Distribute the intersection again
This can be distributed as follows:
\[ (B \cap C') \cup (C \cap C') \]

### Step 8: Simplify further
Since \( C \cap C' = \emptyset \), we have:
\[ B \cap C' \]

### Final Result
Thus, the final simplified expression is:
\[ B \cap C' \]

### Conclusion
The set \( (A \cup B \cup C) \cap (A \cap B' \cap C')' \cap C' \) is equal to \( B \cap C' \).