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 problem, we need to find the Cartesian product of the sets A, B, and C, and then check which option matches the given set of ordered pairs.

### Given Sets:
- \( A = \{ a, b \} \)
- \( B = \{ c, d \} \)
- \( C = \{ d, e \} \)

### Step 1: Find the Cartesian Product \( A \times B \)
The Cartesian product \( A \times B \) is defined as the set of all ordered pairs \( (x, y) \) where \( x \) is from set A and \( y \) is from set B.

\[ A \times B = \{ (a, c), (a, d), (b, c), (b, d) \} \]

### Step 2: Find the Cartesian Product \( A \times C \)
Similarly, we can find the Cartesian product \( A \times C \).

\[ A \times C = \{ (a, d), (a, e), (b, d), (b, e) \} \]

### Step 3: Find the Cartesian Product \( B \times C \)
Now, we find the Cartesian product \( B \times C \).

\[ B \times C = \{ (c, d), (c, e), (d, d), (d, e) \} \]

### Step 4: Combine the Results
Now we need to combine the results from \( A \times B \) and \( A \times C \) to see if they match the given set.

The given set is:
\[ \{ (a, c), (a, d), (a, e), (b, c), (b, d), (b, e) \} \]

### Step 5: Check the Options
Now, we check the options provided in the question. The correct option should match the set we derived.

From the above steps, we see that the set \( \{ (a, c), (a, d), (a, e), (b, c), (b, d), (b, e) \} \) can be represented as \( A \times (B \cup C) \) where \( B \cup C = \{ c, d, e \} \).

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

### Conclusion
The option that matches the given set is \( A \times (B \cup C) \).