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

Options

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

Answer

To solve the problem, we need to simplify the expression \( (A \cup B \cup C) \cap (A \cap B' \cap C')' \).

### Step 1: Understand the components of the expression
The expression consists of two main parts:
1. \( A \cup B \cup C \) - This represents the union of sets A, B, and C.
2. \( (A \cap B' \cap C')' \) - This represents the complement of the intersection of A with the complements of B and C.

### Step 2: Simplify the second part
To simplify \( (A \cap B' \cap C')' \), we can apply De Morgan's Law, which states that the complement of an intersection is the union of the complements. Thus:
\[ (A \cap B' \cap C')' = A' \cup (B')' \cup (C')' = A' \cup B \cup C \]

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

### Step 4: Apply the distributive property
Using the distributive property of sets, we can expand this expression:
\[ = (A \cup B \cup C) \cap (A' \cup B \cup C) = [(A \cap A') \cup (A \cap B) \cup (A \cap C)] \cup [(B \cap A') \cup (B \cap B) \cup (B \cap C)] \cup [(C \cap A') \cup (C \cap B) \cup (C \cap C)] \]

### Step 5: Simplify each part
1. \( A \cap A' = \emptyset \) (the intersection of a set and its complement is empty)
2. \( B \cap B = B \)
3. \( C \cap C = C \)

Thus, the expression simplifies to:
\[ = (A \cap B) \cup (A \cap C) \cup (B \cap A') \cup B \cup (B \cap C) \cup (C \cap A') \cup (C \cap B) \]

### Step 6: Identify the relevant parts
From the above, we can see that the relevant parts that remain after simplification are:
- The union of all elements in B and C, since they cover all intersections with A.

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

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

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