If A,B and C be any three sets , then `A uu (Bnn C)` is the same as :

Options

A. `(A nn C) uu (B nn C) `
B. `(A uu B) nn (A uu C)`
C. `(AuuB) nn (B uu C)`
D. `(A uu C) nn (B uuC)`

Answer

To solve the problem, we need to simplify the expression \( A \cup (B \cap C) \) and find which of the given options is equivalent to it.

### Step-by-Step Solution:

1. **Understand the Expression**: We need to analyze the expression \( A \cup (B \cap C) \). This means we are taking the union of set \( A \) with the intersection of sets \( B \) and \( C \).

2. **Apply Set Operations**: The union operation \( \cup \) combines all elements from both sets, while the intersection operation \( \cap \) includes only the elements that are present in both sets.

3. **Consider the Options**: We have four options to evaluate:
- Option 1: \( A \cap B \cup B \cap C \)
- Option 2: \( A \cup B \cap A \cup C \)
- Option 3: \( A \cup B \cap B \cup C \)
- Option 4: \( A \cup C \cap B \cup C \)

4. **Evaluate Option 2**: Let's simplify \( A \cup (B \cap C) \) and see if it matches with any of the options.
- We can rewrite \( A \cup (B \cap C) \) as \( (A \cup B) \cap (A \cup C) \) using the distributive property of set operations.

5. **Conclusion**: After evaluating, we find that \( A \cup (B \cap C) \) is equivalent to \( A \cup B \cap A \cup C \), which corresponds to Option 2.

### Final Answer:
The correct option is **Option 2: \( A \cup B \cap A \cup C \)**.