If A, B and C are three non-hempty sets such that any two of them are disjoint, then `(AuuBuuC) nn(AnnBnnC)=`

Options

A. A
B. B
C. C
D. `Phi`

Answer

To solve the problem, we need to find the value of the expression \( (A \cup B \cup C) \cap (A \cap B \cap C) \) given that sets A, B, and C are non-empty and any two of them are disjoint.

### Step-by-step Solution:

1. **Understanding Disjoint Sets**:
Since A, B, and C are disjoint, it means that:
- \( A \cap B = \emptyset \)
- \( A \cap C = \emptyset \)
- \( B \cap C = \emptyset \)

This implies that there are no common elements between any two sets.

**Hint**: Recall that disjoint sets have no elements in common.

2. **Finding \( A \cap B \cap C \)**:
Since all pairs of sets are disjoint, the intersection of all three sets \( A \cap B \cap C \) must also be empty. Therefore:
\[ A \cap B \cap C = \emptyset \]

**Hint**: If two sets are disjoint, their intersection is the empty set. Extend this to three sets.

3. **Finding \( A \cup B \cup C \)**:
The union of sets A, B, and C, denoted \( A \cup B \cup C \), contains all elements from A, B, and C. Since they are non-empty, this union is not empty:
\[ A \cup B \cup C \neq \emptyset \]

**Hint**: The union of non-empty sets is non-empty unless all sets are empty.

4. **Calculating the Intersection**:
Now we need to find the intersection of the union with the intersection of the sets:
\[ (A \cup B \cup C) \cap (A \cap B \cap C) = (A \cup B \cup C) \cap \emptyset \]

Since the intersection with the empty set is always empty, we have:
\[ (A \cup B \cup C) \cap \emptyset = \emptyset \]

**Hint**: Remember that the intersection of any set with the empty set is the empty set.

5. **Final Result**:
Therefore, the value of the expression \( (A \cup B \cup C) \cap (A \cap B \cap C) \) is:
\[ \emptyset \]

### Conclusion:
The final answer is:
\[ (A \cup B \cup C) \cap (A \cap B \cap C) = \emptyset \]