If A={1,4},B ={2,3,6} and C= {2,3,7 } then which of the following is correct

Options

A. `A xx (B cup C) = (A xx B) cup (A xx C)`
B. `Axx(B cap C)=(A xxB) cap(AxxC)`
C. `Axx(B-C)=(AxxB)-(AxxC)`
D. `A xx(C-B)=(AxxB)-(A xxC)`

Answer

To solve the problem, we need to analyze the sets A, B, and C and their relationships.

Given:
- A = {1, 4}
- B = {2, 3, 6}
- C = {2, 3, 7}

We need to find out which of the following statements is correct based on the sets provided.

### Step 1: Find B ∪ C
First, we will find the union of sets B and C.

B ∪ C = {2, 3, 6} ∪ {2, 3, 7} = {2, 3, 6, 7}

### Step 2: Find A × B
Next, we will find the Cartesian product of A and B.

A × B = {(1, 2), (1, 3), (1, 6), (4, 2), (4, 3), (4, 6)}

### Step 3: Find A × C
Now, we will find the Cartesian product of A and C.

A × C = {(1, 2), (1, 3), (1, 7), (4, 2), (4, 3), (4, 7)}

### Step 4: Find B - C
Now we will find the difference between sets B and C.

B - C = {2, 3, 6} - {2, 3, 7} = {6}

### Step 5: Find A × (B - C)
Next, we will find the Cartesian product of A and (B - C).

A × (B - C) = A × {6} = {(1, 6), (4, 6)}

### Step 6: Find C - B
Now we will find the difference between sets C and B.

C - B = {2, 3, 7} - {2, 3, 6} = {7}

### Step 7: Find A × (C - B)
Next, we will find the Cartesian product of A and (C - B).

A × (C - B) = A × {7} = {(1, 7), (4, 7)}

### Step 8: Find the union of results from Step 5 and Step 7
Now we will find the union of the results from Step 5 and Step 7.

Result = A × (B - C) ∪ A × (C - B) = {(1, 6), (4, 6)} ∪ {(1, 7), (4, 7)} = {(1, 6), (4, 6), (1, 7), (4, 7)}

### Step 9: Check the correctness of the options
We now need to check which of the provided options is correct based on our findings.

1. **Option 1**: A × B ∪ A × C = {(1, 2), (1, 3), (1, 6), (4, 2), (4, 3), (4, 6)} ∪ {(1, 2), (1, 3), (1, 7), (4, 2), (4, 3), (4, 7)} = {(1, 2), (1, 3), (1, 6), (1, 7), (4, 2), (4, 3), (4, 6), (4, 7)}. This option is correct.

2. **Option 2**: B ∩ C = {2, 3} and A × (B ∩ C) = {(1, 2), (1, 3), (4, 2), (4, 3)}. This option is also correct.

3. **Option 3**: A × (B - C) = {(1, 6), (4, 6)} and A × (C - B) = {(1, 7), (4, 7)}. The union of these two is {(1, 6), (4, 6), (1, 7), (4, 7)}, which does not match the expected result. This option is incorrect.

### Conclusion
The correct options are 1 and 2.