If A, B and C are three finite sets, then what is `[(A uu B) nn C]'` equal to?

Options

A. `A' uu B' nn C'`
B. `A' nn B' nn C'`
C. `A' nn B' uu C'`
D. `A nn B nn C`

Answer

To solve the problem, we need to find the expression \([(A \cup B) \cap C]'\), where \(A\), \(B\), and \(C\) are three finite sets. The prime symbol (') denotes the complement of the set.

### Step-by-step Solution:

1. **Understanding the Expression**:
We start with the expression \([(A \cup B) \cap C]'\). This means we first need to find the union of sets \(A\) and \(B\), then intersect that result with set \(C\), and finally take the complement of the resulting set.

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

3. **Finding the Complement of the Union**:
Now we need to find the complement of the union \(A \cup B\). Again, using De Morgan's Laws, we have:
\[ (A \cup B)' = A' \cap B' \]

4. **Combine the Results**:
Substituting back into our expression, we get:
\[ (A \cup B)' \cup C' = (A' \cap B') \cup C' \]

5. **Final Expression**:
Therefore, the final expression for \([(A \cup B) \cap C]'\) is:
\[ (A' \cap B') \cup C' \]

### Conclusion:
The expression \([(A \cup B) \cap C]'\) simplifies to \((A' \cap B') \cup C'\).