If A B, C are any three sets, then :

Options

A. `A-(BuuC)=(A-B)uu(A-C)`
B. `A-(BuuC)=(A-B)nn(A-C)`
C. `A-(BnnC)=(A-B)uu(A-C)`
D. `A-(BuuC)=(AuuB)-(AuuC)`

Answer

To solve the problem, we need to determine which of the given options is true based on the properties of sets A, B, and C. Let's denote the sets as follows:

- Set A = {1, 2, 3, 4}
- Set B = {4, 5, 6, 7}
- Set C = {8, 9, 10, 11}

We will evaluate each option step by step.

### Step 1: Evaluate Option A
The expression for Option A is:
\[ A - (B \cup C) = (A - B) \cup (A - C) \]

1. Calculate \( B \cup C \):
\[ B \cup C = \{4, 5, 6, 7\} \cup \{8, 9, 10, 11\} = \{4, 5, 6, 7, 8, 9, 10, 11\} \]

2. Calculate \( A - (B \cup C) \):
\[ A - (B \cup C) = \{1, 2, 3, 4\} - \{4, 5, 6, 7, 8, 9, 10, 11\} = \{1, 2, 3\} \]

3. Calculate \( A - B \):
\[ A - B = \{1, 2, 3, 4\} - \{4, 5, 6, 7\} = \{1, 2, 3\} \]

4. Calculate \( A - C \):
\[ A - C = \{1, 2, 3, 4\} - \{8, 9, 10, 11\} = \{1, 2, 3, 4\} \]

5. Now calculate \( (A - B) \cup (A - C) \):
\[ (A - B) \cup (A - C) = \{1, 2, 3\} \cup \{1, 2, 3, 4\} = \{1, 2, 3, 4\} \]

6. Compare both sides:
\[ A - (B \cup C) = \{1, 2, 3\} \quad \text{and} \quad (A - B) \cup (A - C) = \{1, 2, 3, 4\} \]
They are not equal, so Option A is incorrect.

### Step 2: Evaluate Option B
The expression for Option B is:
\[ A - (B \cap C) = (A - B) \cup (A - C) \]

1. Calculate \( B \cap C \):
\[ B \cap C = \{4, 5, 6, 7\} \cap \{8, 9, 10, 11\} = \emptyset \]

2. Calculate \( A - (B \cap C) \):
\[ A - (B \cap C) = \{1, 2, 3, 4\} - \emptyset = \{1, 2, 3, 4\} \]

3. We already calculated \( A - B \) and \( A - C \):
\[ A - B = \{1, 2, 3\} \quad \text{and} \quad A - C = \{1, 2, 3, 4\} \]

4. Now calculate \( (A - B) \cup (A - C) \):
\[ (A - B) \cup (A - C) = \{1, 2, 3\} \cup \{1, 2, 3, 4\} = \{1, 2, 3, 4\} \]

5. Compare both sides:
\[ A - (B \cap C) = \{1, 2, 3, 4\} \quad \text{and} \quad (A - B) \cup (A - C) = \{1, 2, 3, 4\} \]
They are equal, so Option B is correct.

### Conclusion
The correct option is **B**.