Let `A={1,2},B={2,3,4},C ={4,5},`find
`A times (B cap C)`

Answer

To solve the problem of finding \( A \times (B \cap C) \), we will follow these steps:

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

### Step 2: Find the intersection of sets B and C
The intersection \( B \cap C \) consists of elements that are common to both sets \( B \) and \( C \).
- Elements in \( B \) are \( 2, 3, 4 \)
- Elements in \( C \) are \( 4, 5 \)

Now, we check which elements are common:
- \( 2 \) is not in \( C \)
- \( 3 \) is not in \( C \)
- \( 4 \) is in both \( B \) and \( C \)

Thus, the intersection is:
\[ B \cap C = \{4\} \]

### Step 3: Find the Cartesian product \( A \times (B \cap C) \)
Now we need to find \( A \times (B \cap C) \), which means we will create ordered pairs from each element in \( A \) with each element in \( B \cap C \).

Since \( B \cap C = \{4\} \), we will pair each element in \( A \) with \( 4 \):
- Pair \( 1 \) from \( A \) with \( 4 \): \( (1, 4) \)
- Pair \( 2 \) from \( A \) with \( 4 \): \( (2, 4) \)

Thus, the Cartesian product is:
\[ A \times (B \cap C) = \{(1, 4), (2, 4)\} \]

### Final Answer
The result is:
\[ A \times (B \cap C) = \{(1, 4), (2, 4)\} \]