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

Options

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

Answer

To solve the set expression \( (A \cup B \cup C) \cap (A \cap B' \cap C')' \cap C \), we will follow the steps outlined in the video transcript and apply set theory principles, particularly De Morgan's laws.

### Step-by-Step Solution:

1. **Write the original expression**:
\[ (A \cup B \cup C) \cap (A \cap B' \cap C')' \cap C \]

2. **Apply De Morgan's Law**:
According to De Morgan's Law, the complement of an intersection is the union of the complements. Thus, we can rewrite \( (A \cap B' \cap C')' \) as:
\[ (A \cap B' \cap C')' = A' \cup (B')' \cup (C')' = A' \cup B \cup C \]

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

4. **Simplify the expression**:
We can simplify the intersection \( (A \cup B \cup C) \cap (A' \cup B \cup C) \):
- The intersection of \( A \) and \( A' \) is empty (\( A \cap A' = \emptyset \)).
- Therefore, we focus on the remaining parts:
\[ (B \cup C) \cap (A' \cup B \cup C) = (B \cup C) \]

5. **Combine with the last part of the expression**:
Now we need to intersect \( (B \cup C) \) with \( C \):
\[ (B \cup C) \cap C \]

6. **Final simplification**:
The intersection of \( (B \cup C) \) with \( C \) gives us:
\[ C \]

Thus, the final result is:
\[ C \]

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