If A,B and C are three non=empty sets such that A and B are disjoint and the number of elements contained in A is equal to those contained in the set of elements common to the set A and C, then `n(AcupBcupC)` is necessarily equal to

Options

A. `n(BuuC)`
B. `n(AuuC)`
C. both a and b
D. none of these

Answer

To solve the problem step by step, we will analyze the given information about the sets A, B, and C.

### Step 1: Understand the properties of the sets
- We are given that sets A and B are disjoint. This means that they have no elements in common, which can be mathematically expressed as:
\[ A \cap B = \emptyset \]
- We also know that the number of elements in set A is equal to the number of elements that are common to sets A and C. This can be expressed as:
\[ n(A) = n(A \cap C) \]

### Step 2: Analyze the implications of the given conditions
- The condition \( n(A) = n(A \cap C) \) implies that all elements of A are also in C. Therefore, we can conclude that:
\[ A \subseteq C \]
- Since A is a subset of C, when we take the union of A and C, we have:
\[ A \cup C = C \]

### Step 3: Calculate the union of all three sets
- We need to find \( n(A \cup B \cup C) \). Using the property we derived in Step 2, we can rewrite this as:
\[ n(A \cup B \cup C) = n(B \cup (A \cup C)) = n(B \cup C) \]
- Since A and B are disjoint, the number of elements in the union of B and C does not include any elements from A. Thus, we can express this as:
\[ n(A \cup B \cup C) = n(B) + n(C) \]

### Step 4: Conclusion
- Since we have established that \( n(A \cup B \cup C) = n(B \cup C) \), we conclude that:
\[ n(A \cup B \cup C) = n(B) + n(C) \]
- Therefore, the final answer is:
\[ n(A \cup B \cup C) = n(B \cup C) \]

### Summary
The number of elements in the union of sets A, B, and C is equal to the number of elements in the union of sets B and C.