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 determine which of the given options is equal to the set `{(a,c),(a,d),(a,e),(b,c),(b,d),(b,e)}`.

Let's denote the sets:
- \( A = \{a, b\} \)
- \( B = \{c, d\} \)
- \( C = \{d, e\} \)

Now, we will evaluate each option step by step.

### Step 1: Evaluate Option 1 - \( A \cup (B \cap C) \)
1. **Find \( B \cap C \)**:
- \( B = \{c, d\} \)
- \( C = \{d, e\} \)
- The intersection \( B \cap C = \{d\} \) (only the element 'd' is common).

2. **Find \( A \cup (B \cap C) \)**:
- \( A = \{a, b\} \)
- So, \( A \cup \{d\} = \{a, b, d\} \).

3. **Conclusion**:
- This does not match our set `{(a,c),(a,d),(a,e),(b,c),(b,d),(b,e)}`.

### Step 2: Evaluate Option 2 - \( A \cup (B \cap C) \)
1. **Find \( B \cap C \)**:
- As calculated before, \( B \cap C = \{d\} \).

2. **Find \( A \cup (B \cap C) \)**:
- \( A \cup \{d\} = \{a, b, d\} \).

3. **Conclusion**:
- This does not match our set.

### Step 3: Evaluate Option 3 - \( A \times (B \cup C) \)
1. **Find \( B \cup C \)**:
- \( B = \{c, d\} \)
- \( C = \{d, e\} \)
- The union \( B \cup C = \{c, d, e\} \) (combine all elements, but 'd' is counted only once).

2. **Find \( A \times (B \cup C) \)**:
- \( A = \{a, b\} \)
- So, \( A \times (B \cup C) = \{(a,c), (a,d), (a,e), (b,c), (b,d), (b,e)\} \).

3. **Conclusion**:
- This matches our set `{(a,c),(a,d),(a,e),(b,c),(b,d),(b,e)}`.

### Step 4: Evaluate Option 4
- Since we have already found that Option 3 is correct, we do not need to check Option 4.

### Final Answer
The correct option that is equal to the set `{(a,c),(a,d),(a,e),(b,c),(b,d),(b,e)}` is **Option 3: \( A \times (B \cup C) \)**.