If `A={2,3},B={4,5}andC={5,6}`, then `n{(AxxB)uu(BxxC)}` is

Explanation

To solve the problem, we need to find the number of elements in the union of the Cartesian products \( A \times B \) and \( B \times C \).

**Step 1: Identify the sets**
- Given sets are:
- \( A = \{2, 3\} \)
- \( B = \{4, 5\} \)
- \( C = \{5, 6\} \)

**Step 2: Calculate the Cartesian product \( A \times B \)**
- The Cartesian product \( A \times B \) consists of all ordered pairs where the first element is from \( A \) and the second element is from \( B \).
- Thus, we calculate:
\[ A \times B = \{(2, 4), (2, 5), (3, 4), (3, 5)\} \]

**Step 3: Calculate the Cartesian product \( B \times C \)**
- The Cartesian product \( B \times C \) consists of all ordered pairs where the first element is from \( B \) and the second element is from \( C \).
- Thus, we calculate:
\[ B \times C = \{(4, 5), (4, 6), (5, 5), (5, 6)\} \]

**Step 4: Find the union \( (A \times B) \cup (B \times C) \)**
- Now, we need to find the union of the two sets obtained in steps 2 and 3.
- Combine the elements:
\[ (A \times B) \cup (B \times C) = \{(2, 4), (2, 5), (3, 4), (3, 5)\} \cup \{(4, 5), (4, 6), (5, 5), (5, 6)\} \]
- Listing all elements together:
\[ = \{(2, 4), (2, 5), (3, 4), (3, 5), (4, 5), (4, 6), (5, 5), (5, 6)\} \]

**Step 5: Count the number of distinct elements**
- Now, we count the distinct ordered pairs in the union:
- The distinct pairs are:
1. \( (2, 4) \)
2. \( (2, 5) \)
3. \( (3, 4) \)
4. \( (3, 5) \)
5. \( (4, 5) \)
6. \( (4, 6) \)
7. \( (5, 5) \)
8. \( (5, 6) \)
- Therefore, the number of elements \( n((A \times B) \cup (B \times C)) = 8 \).

**Final Answer:**
\[ n((A \times B) \cup (B \times C)) = 8 \]

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