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 and then determine the number of subsets of the Cartesian product A × B. Let's break it down step by step.

### Step 1: Determine the Set A
We have the set defined as:
\[ A = \{ x \in \mathbb{Z} : |x - 3|^{(x^2 - 5x + 6)} = 1 \} \]

To analyze this, we can break it down into cases based on the properties of exponents and absolute values.

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

2. **Case 2:** \( x^2 - 5x + 6 = 0 \)
- Factoring gives us:
\[ (x - 2)(x - 3) = 0 \]
- This results in:
- \( x = 2 \)
- \( x = 3 \) (but we will discard this since it leads to zero in the exponent)

Since \( |x - 3|^{(x^2 - 5x + 6)} = 1 \) can also hold if the exponent is zero (which we already considered), we conclude that the valid integers for set A are:
\[ A = \{2, 4\} \]

### Step 2: Determine the Set B
The set B is defined as:
\[ B = \{ x \in \mathbb{Z} : 10 < 3x + 1 < 22 \} \]

We can solve this compound inequality:
1. From \( 3x + 1 > 10 \):
\[ 3x > 9 \]
\[ x > 3 \]

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

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

The integers satisfying this condition are:
\[ B = \{4, 5, 6\} \]

### Step 3: Calculate the Number of Elements in A and B
- The number of elements in set A:
\[ |A| = 2 \]
- The number of elements in set B:
\[ |B| = 3 \]

### Step 4: Calculate the Number of Elements in A × B
The number of elements in the Cartesian product \( A \times B \) is given by:
\[ |A \times B| = |A| \times |B| = 2 \times 3 = 6 \]

### Step 5: 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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