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 start by defining the sets given in the problem:
- \( A = \{1, 2, 3\} \)
- \( B = \{3, 4\} \)
- \( C = \{4, 5, 6\} \)
### 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 will take the union of the two sets calculated above:
\[
(A \times B) \cup (A \times C) = \{(1,3), (1,4), (2,3), (2,4), (3,3), (3,4)\} \cup \{(1,4), (1,5), (1,6), (2,4), (2,5), (2,6), (3,4), (3,5), (3,6)\}
\]
### Step 5: Combine and Remove Duplicates
Combining the two sets and removing duplicates:
- From \( A \times B \): \( (1,3), (1,4), (2,3), (2,4), (3,3), (3,4) \)
- From \( A \times C \): \( (1,5), (1,6), (2,5), (2,6), (3,5), (3,6) \)
The combined set is:
\[
\{(1,3), (1,4), (2,3), (2,4), (3,3), (3,4), (1,5), (1,6), (2,5), (2,6), (3,5), (3,6)\}
\]
### Final Result
Thus, we have:
\[
(A \times B) \cup (A \times C) = \{(1,3), (1,4), (1,5), (1,6), (2,3), (2,4), (2,5), (2,6), (3,3), (3,4), (3,5), (3,6)\}
\]
This matches the set given in the problem statement.
