The set `(AuuBuuC)nn(AnnB'nnC')'` is equal to

Options

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

Answer

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

### Step 1: Understand the Components
We have two main parts in the expression:
1. \( A \cup B \cup C \)
2. \( (A \cap B' \cap C')' \)

### Step 2: Simplify the Second Part
The second part \( (A \cap B' \cap C')' \) represents the complement of the intersection of \( A \), the complement of \( B \), and the complement of \( C \). By De Morgan's laws, we can express this as:
\[ (A \cap B' \cap C')' = A' \cup B \cup C \]

### Step 3: Combine the Two Parts
Now we need to find the intersection of the two parts:
\[ (A \cup B \cup C) \cap (A' \cup B \cup C) \]

### Step 4: Apply the Distributive Law
Using the distributive property of sets, we can rewrite the intersection:
\[ = (A \cup B \cup C) \cap (A' \cup B \cup C) = ((A \cup B \cup C) \cap A') \cup ((A \cup B \cup C) \cap B) \cup ((A \cup B \cup C) \cap C) \]

### Step 5: Evaluate Each Intersection
1. \( (A \cup B \cup C) \cap A' \) gives us the elements in \( B \cup C \) that are not in \( A \).
2. \( (A \cup B \cup C) \cap B \) gives us the elements in \( B \).
3. \( (A \cup B \cup C) \cap C \) gives us the elements in \( C \).

### Step 6: Combine the Results
From the evaluations, we can see that the intersection simplifies to:
\[ B \cup C \]

### Final Result
Thus, the set \( (A \cup B \cup C) \cap (A \cap B' \cap C')' \) is equal to:
\[ B \cup C \]