If `A={a,b},B={c,d},C={d,e}`, then `{(a,c),(a,d),(a,e),(b,c),(b,d),(b,e)}` is equal to

Options

A. `Ann(BuuC)`
B. `Auu(BnnC)`
C. `Axx(BuuC)`
D. `Axx(BnnC)`

Explanation

To solve the given problem, we need to find the Cartesian product of the set \( A \) with the union of the sets \( B \) and \( C \).

1. **Identify the sets**:
- \( A = \{a, b\} \)
- \( B = \{c, d\} \)
- \( C = \{d, e\} \)

2. **Find the union of sets \( B \) and \( C \)**:
\[ B \cup C = \{c, d\} \cup \{d, e\} \]
Combining the elements, we get:
\[ B \cup C = \{c, d, e\} \]

3. **Find the Cartesian product \( A \times (B \cup C) \)**:
The Cartesian product \( A \times (B \cup C) \) is defined as the set of all ordered pairs \( (x, y) \) where \( x \) is from set \( A \) and \( y \) is from set \( B \cup C \).
\[ A \times (B \cup C) = \{(a, c), (a, d), (a, e), (b, c), (b, d), (b, e)\} \]

4. **Conclusion**:
Therefore, the set \( \{(a,c),(a,d),(a,e),(b,c),(b,d),(b,e)\} \) is equal to \( A \times (B \cup C) \).

So the final answer is:
\[ \{(a,c),(a,d),(a,e),(b,c),(b,d),(b,e)\} = A \times (B \cup C) \]

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