If `A={1,2,3},B={3,4}and C={4,5,6}, "then prove that" (AxxB)uu(AxxC)` `={(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5 ),(2,6),(3,3),(3,4),(3,5),(3,6)}`

Explanation

To prove that \( (A \times B) \cup (A \times C) = \{(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,3),(3,4),(3,5),(3,6)\} \), we will follow these steps:

### Step 1: Define the Sets
We have the following sets:
- \( A = \{1, 2, 3\} \)
- \( B = \{3, 4\} \)
- \( C = \{4, 5, 6\} \)

### Step 2: Calculate \( A \times B \)
The Cartesian product \( A \times B \) consists of all ordered pairs \( (a, b) \) where \( a \in A \) and \( b \in B \).

- For \( a = 1 \):
- \( (1, 3) \)
- \( (1, 4) \)

- For \( a = 2 \):
- \( (2, 3) \)
- \( (2, 4) \)

- For \( a = 3 \):
- \( (3, 3) \)
- \( (3, 4) \)

Thus,
\[ A \times B = \{(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)\} \]

### Step 3: Calculate \( A \times C \)
Next, we calculate \( A \times C \) which consists of all ordered pairs \( (a, c) \) where \( a \in A \) and \( c \in C \).

- For \( a = 1 \):
- \( (1, 4) \)
- \( (1, 5) \)
- \( (1, 6) \)

- For \( a = 2 \):
- \( (2, 4) \)
- \( (2, 5) \)
- \( (2, 6) \)

- For \( a = 3 \):
- \( (3, 4) \)
- \( (3, 5) \)
- \( (3, 6) \)

Thus,
\[ A \times C = \{(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)\} \]

### Step 4: Calculate \( (A \times B) \cup (A \times C) \)
Now we take the union of \( A \times B \) and \( A \times C \).

From \( A \times B \):
\[ \{(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)\} \]

From \( A \times C \):
\[ \{(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)\} \]

Now, we combine these sets:
- Include \( (1, 3) \)
- Include \( (1, 4) \) (already included)
- Include \( (2, 3) \)
- Include \( (2, 4) \) (already included)
- Include \( (3, 3) \)
- Include \( (3, 4) \) (already included)
- Include \( (1, 5) \)
- Include \( (1, 6) \)
- Include \( (2, 5) \)
- Include \( (2, 6) \)
- Include \( (3, 5) \)
- Include \( (3, 6) \)

Thus, the union is:
\[ (A \times B) \cup (A \times C) = \{(1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6)\} \]

### Conclusion
We have shown that:
\[ (A \times B) \cup (A \times C) = \{(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,3),(3,4),(3,5),(3,6)\} \]