Given the sets $A = \{1, 2, 3\}$, $B = \{3, 4\}$, and $C = \{4, 5, 6\}$, we are tasked with finding $(A \times B) \cap (A \times C)$, where $A \times B$ denotes the Cartesian product of sets $A$ and $B$.

Step 1: Find $A \times B$

The Cartesian product $A \times B$ consists of all ordered pairs where the first element comes from $A$ and the second from $B$. Thus, we compute:

$A \times B = \{ (1,3), (1,4), (2,3), (2,4), (3,3), (3,4) \}$

Step 2: Find $A \times C$

Similarly, $A \times C$ consists of all ordered pairs where the first element comes from $A$ and the second from $C$:

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

Step 3: Find $(A \times B) \cap (A \times C)$

Now, we find the intersection of $A \times B$ and $A \times C$, which are the pairs that appear in both sets.

$(A \times B) \cap (A \times C) = \{ (1,4), (2,4), (3,4) \}$

Final Answer:

$(A \times B) \cap (A \times C) = \{ (1,4), (2,4), (3,4) \}$

Would you like any details or further explanation?

Here are 5 related questions:

1. How is the Cartesian product of two sets defined?
2. What are some properties of the Cartesian product?
3. How do you find the intersection of two sets in general?
4. How would the result change if $B$ and $C$ were different?
5. Can the Cartesian product operation be extended to more than two sets?

Tip: When finding intersections, it's often useful to list out elements explicitly to avoid missing any common pairs.