If `A = {1,2,3}, B = {3,4} , C = {4,5,6}`, then number of elements in the set `(AxxB)nn(BxxC)` is equal to

Options

A. 1
B. 2
C. 3
D. 0

Explanation

To solve the problem, we need to find the number of elements in the set \( (A \times B) \cap (B \times C) \).

### Step 1: Determine the Cartesian Product \( A \times B \)

Given:
- \( A = \{1, 2, 3\} \)
- \( B = \{3, 4\} \)

The Cartesian product \( A \times B \) consists of all ordered pairs where the first element is from set \( A \) and the second element is from set \( B \).

\[ A \times B = \{(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)\} \]

### Step 2: Determine the Cartesian Product \( B \times C \)

Given:
- \( B = \{3, 4\} \)
- \( C = \{4, 5, 6\} \)

The Cartesian product \( B \times C \) consists of all ordered pairs where the first element is from set \( B \) and the second element is from set \( C \).

\[ B \times C = \{(3, 4), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6)\} \]

### Step 3: Find the Intersection \( (A \times B) \cap (B \times C) \)

Now, we need to find the intersection of the two sets \( A \times B \) and \( B \times C \). This means we are looking for common ordered pairs in both sets.

- From \( A \times B \): \( \{(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)\} \)
- From \( B \times C \): \( \{(3, 4), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6)\} \)

Now, let's check each pair from \( A \times B \) to see if it exists in \( B \times C \):

- \( (1, 3) \) is not in \( B \times C \)
- \( (1, 4) \) is not in \( B \times C \)
- \( (2, 3) \) is not in \( B \times C \)
- \( (2, 4) \) is not in \( B \times C \)
- \( (3, 3) \) is not in \( B \times C \)
- \( (3, 4) \) **is in** \( B \times C \)

Thus, the intersection \( (A \times B) \cap (B \times C) \) is:

\[ (A \times B) \cap (B \times C) = \{(3, 4)\} \]

### Step 4: Count the Number of Elements in the Intersection

The number of elements in the intersection set \( (A \times B) \cap (B \times C) \) is:

\[ \text{Number of elements} = 1 \]

### Final Answer

The number of elements in the set \( (A \times B) \cap (B \times C) \) is **1**.

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