Let Z is be the set of integers , if `A={"x"inZ:|x-3|^((x^2-5x+6))=1} and B{x in Z : 10 lt3x+1lt 22}`, then the number of subsets of the set `AxxB` is

Options

A. `2^6`
B. `2^8`
C. `2^15`
D. `2^9`

Answer

To solve the problem, we need to find the sets A and B based on the given conditions and then determine the number of subsets of the Cartesian product A × B.

### Step 1: Determine the set A

We are given the condition for set A as:
\[ A = \{ x \in \mathbb{Z} : |x - 3|^{(x^2 - 5x + 6)} = 1 \} \]

The expression \( |x - 3|^{(x^2 - 5x + 6)} = 1 \) can be true under two conditions:
1. \( |x - 3| = 1 \)
2. \( x^2 - 5x + 6 = 0 \)

#### Case 1: \( |x - 3| = 1 \)

This gives us two equations:
1. \( x - 3 = 1 \) → \( x = 4 \)
2. \( x - 3 = -1 \) → \( x = 2 \)

So from this case, we get \( x = 2 \) and \( x = 4 \).

#### Case 2: \( x^2 - 5x + 6 = 0 \)

Factoring the quadratic:
\[ (x - 2)(x - 3) = 0 \]
This gives us:
1. \( x = 2 \)
2. \( x = 3 \)

However, we need to ensure that \( |x - 3|^{(x^2 - 5x + 6)} \) does not equal 0. If \( x = 3 \), the exponent becomes 0, which makes the expression equal to 1, but we want to exclude this case since \( |x - 3| \) would be 0.

Thus, from this case, we only take \( x = 2 \).

### Conclusion for set A:
Combining both cases, we have:
\[ A = \{ 2, 4 \} \]

### Step 2: Determine the set B

We are given the condition for set B as:
\[ B = \{ x \in \mathbb{Z} : 10 < 3x + 1 < 22 \} \]

We can break this down into two inequalities:
1. \( 3x + 1 > 10 \)
2. \( 3x + 1 < 22 \)

#### Solving the inequalities:

1. From \( 3x + 1 > 10 \):
\[ 3x > 9 \]
\[ x > 3 \]

2. From \( 3x + 1 < 22 \):
\[ 3x < 21 \]
\[ x < 7 \]

Combining these results, we have:
\[ 3 < x < 7 \]

The integer solutions for \( x \) are:
\[ x = 4, 5, 6 \]

### Conclusion for set B:
Thus, we have:
\[ B = \{ 4, 5, 6 \} \]

### Step 3: Find the Cartesian product A × B

Now we need to find the Cartesian product \( A \times B \):
- Set A has 2 elements: \( |A| = 2 \)
- Set B has 3 elements: \( |B| = 3 \)

The number of elements in the Cartesian product \( A \times B \) is:
\[ |A \times B| = |A| \times |B| = 2 \times 3 = 6 \]

### Step 4: Calculate the number of subsets of A × B

The number of subsets of a set with \( n \) elements is given by \( 2^n \). Therefore, the number of subsets of \( A \times B \) is:
\[ \text{Number of subsets} = 2^{|A \times B|} = 2^6 = 64 \]

### Final Answer:
The number of subsets of the set \( A \times B \) is \( 64 \).

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