If `A={x:x^(2)-5x+6=0},B={2,4},C={4,5}` then find `Axx(BnnC)`

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Find the Set A
We need to solve the equation \(x^2 - 5x + 6 = 0\).

1. Factor the quadratic equation:
\[ x^2 - 5x + 6 = (x - 2)(x - 3) = 0 \]
2. Set each factor to zero:
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]
3. Thus, the set \(A\) is:
\[ A = \{2, 3\} \]

### Step 2: Identify Sets B and C
We are given:
- \(B = \{2, 4\}\)
- \(C = \{4, 5\}\)

### Step 3: Find the Intersection of Sets B and C
We need to find \(B \cap C\):
- The elements in set \(B\) are \(2\) and \(4\).
- The elements in set \(C\) are \(4\) and \(5\).
- The common element in both sets is \(4\).

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

### Step 4: Find the Cartesian Product \(A \times (B \cap C)\)
Now we need to find \(A \times (B \cap C)\):
- Set \(A = \{2, 3\}\)
- Set \(B \cap C = \{4\}\)

The Cartesian product \(A \times (B \cap C)\) is given by pairing each element of \(A\) with each element of \(B \cap C\):
\[ A \times (B \cap C) = \{(2, 4), (3, 4)\} \]

### Final Answer
Thus, the final answer is:
\[ A \times (B \cap C) = \{(2, 4), (3, 4)\} \]

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