Given `A={1,2),B={3),C={4,5}`, test whether the following are true :
`Axx(BcupC)=(AcupB)xx(AcupC)`.

Answer

To determine whether the statement \( A \times (B \cup C) = (A \cup B) \times (A \cup C) \) is true, we will evaluate both sides of the equation step by step.

### Step 1: Define the Sets
Given:
- \( A = \{1, 2\} \)
- \( B = \{3\} \)
- \( C = \{4, 5\} \)

### Step 2: Calculate \( B \cup C \)
The union of sets \( B \) and \( C \) is:
\[ B \cup C = \{3\} \cup \{4, 5\} = \{3, 4, 5\} \]

### Step 3: Calculate \( A \times (B \cup C) \)
Now we compute the Cartesian product \( A \times (B \cup C) \):
\[ A \times (B \cup C) = A \times \{3, 4, 5\} \]
Calculating this:
- Pairing each element of \( A \) with each element of \( B \cup C \):
\[ A \times (B \cup C) = \{(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)\} \]

### Step 4: Calculate \( A \cup B \) and \( A \cup C \)
Next, we find the unions:
\[ A \cup B = \{1, 2\} \cup \{3\} = \{1, 2, 3\} \]
\[ A \cup C = \{1, 2\} \cup \{4, 5\} = \{1, 2, 4, 5\} \]

### Step 5: Calculate \( (A \cup B) \times (A \cup C) \)
Now we compute the Cartesian product \( (A \cup B) \times (A \cup C) \):
\[ (A \cup B) \times (A \cup C) = \{1, 2, 3\} \times \{1, 2, 4, 5\} \]
Calculating this:
- Pairing each element of \( A \cup B \) with each element of \( A \cup C \):
\[ (A \cup B) \times (A \cup C) = \{(1, 1), (1, 2), (1, 4), (1, 5), (2, 1), (2, 2), (2, 4), (2, 5), (3, 1), (3, 2), (3, 4), (3, 5)\} \]

### Step 6: Compare Both Sides
Now we compare the results from Step 3 and Step 5:
- Left Hand Side:
\[ A \times (B \cup C) = \{(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)\} \]
- Right Hand Side:
\[ (A \cup B) \times (A \cup C) = \{(1, 1), (1, 2), (1, 4), (1, 5), (2, 1), (2, 2), (2, 4), (2, 5), (3, 1), (3, 2), (3, 4), (3, 5)\} \]

### Conclusion
Since the left-hand side is not equal to the right-hand side, we conclude that:
\[ A \times (B \cup C) \neq (A \cup B) \times (A \cup C) \]
Thus, the statement is **not true**.