If `A={1,2,3},B={4,5,6} and C={1,2},` then: `(A-B)xx(AcapC)=`

Options

A. `{(1,3),(1,5)}`
B. `{(2,1),(2,2),(2,3)}`
C. `{(1,2),(1,3),(1,5)}`
D. none of these.

Answer

To solve the problem, we need to find the value of \( (A - B) \times (A \cap C) \) given the sets \( A = \{1, 2, 3\} \), \( B = \{4, 5, 6\} \), and \( C = \{1, 2\} \).

### Step 1: Calculate \( A - B \)

The difference \( A - B \) means we take all elements from set \( A \) that are not in set \( B \).

- Elements of \( A \): \( \{1, 2, 3\} \)
- Elements of \( B \): \( \{4, 5, 6\} \)

Since none of the elements in \( A \) are present in \( B \), we have:

\[ A - B = \{1, 2, 3\} \]

### Step 2: Calculate \( A \cap C \)

The intersection \( A \cap C \) means we take all elements that are common to both sets \( A \) and \( C \).

- Elements of \( A \): \( \{1, 2, 3\} \)
- Elements of \( C \): \( \{1, 2\} \)

The common elements are:

\[ A \cap C = \{1, 2\} \]

### Step 3: Calculate \( (A - B) \times (A \cap C) \)

Now we need to find the Cartesian product \( (A - B) \times (A \cap C) \).

- From Step 1, \( A - B = \{1, 2, 3\} \)
- From Step 2, \( A \cap C = \{1, 2\} \)

The Cartesian product \( X \times Y \) is defined as the set of all ordered pairs \( (x, y) \) where \( x \) is from set \( X \) and \( y \) is from set \( Y \).

Thus, we compute:

\[ (A - B) \times (A \cap C) = \{1, 2, 3\} \times \{1, 2\} \]

This gives us the following pairs:

- For \( 1 \): \( (1, 1), (1, 2) \)
- For \( 2 \): \( (2, 1), (2, 2) \)
- For \( 3 \): \( (3, 1), (3, 2) \)

So, the complete set of ordered pairs is:

\[ (A - B) \times (A \cap C) = \{(1, 1), (1, 2), (2, 1), (2, 2), (3, 1), (3, 2)\} \]

### Final Answer

Thus, the final answer is:

\[ (A - B) \times (A \cap C) = \{(1, 1), (1, 2), (2, 1), (2, 2), (3, 1), (3, 2)\} \]

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