The set `Ann(Buu(B'nnC)uu(B'nnC'))` is equal to (i) `BnnC` (ii) `BnnC'` (iii) `A` (iv) `B`

Options

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

Answer

To solve the question, we need to simplify the expression \( A \cap (B \cup (B' \cap C) \cup (B' \cap C')) \) step by step.

### Step 1: Write the expression
We start with the expression:
\[ A \cap (B \cup (B' \cap C) \cup (B' \cap C')) \]

### Step 2: Apply the distributive law
Using the distributive law, we can rearrange the terms:
\[ A \cap (B \cup (B' \cap (C \cup C'))) \]

### Step 3: Simplify \( C \cup C' \)
Since \( C \cup C' \) is the universal set (or the entire space), we can simplify this to:
\[ A \cap (B \cup U) \]
where \( U \) is the universal set.

### Step 4: Apply the property of intersection with the universal set
The intersection of any set with the universal set is the set itself. Therefore:
\[ A \cap (B \cup U) = A \]

### Conclusion
Thus, the expression simplifies to:
\[ A \]

### Final Answer
The set \( A \cap (B \cup (B' \cap C) \cup (B' \cap C')) \) is equal to \( A \), which corresponds to option (iii) \( A \).

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