Let Z be the set of integers. If A = `{x in Z : 2^((x + 2)(x^(2) - 5x + 6)} = 1` and `B = {x in Z : -3 lt 2x - 1 lt 9}`, then the number of subsets of the set A `xx` B is

Options

A. `2^(18)`
B. `2^(10)`
C. `2^(15)`
D. `2^(12)`

Answer

To solve the given problem, we need to determine the sets A and B first, and then calculate the number of subsets of the Cartesian product of these two sets.

### Step 1: Determine the set A

The set A is defined as:
\[ A = \{ x \in \mathbb{Z} : 2^{(x + 2)(x^2 - 5x + 6)} = 1 \} \]

For the equation \( 2^{(x + 2)(x^2 - 5x + 6)} = 1 \) to hold, the exponent must equal 0 (since \( 2^0 = 1 \)). Therefore, we set:
\[ (x + 2)(x^2 - 5x + 6) = 0 \]

This gives us two factors to consider:
1. \( x + 2 = 0 \)
2. \( x^2 - 5x + 6 = 0 \)

**Solving the first factor:**
\[ x + 2 = 0 \Rightarrow x = -2 \]

**Solving the second factor:**
We can factor the quadratic:
\[ x^2 - 5x + 6 = (x - 2)(x - 3) = 0 \]
This gives us:
\[ x - 2 = 0 \Rightarrow x = 2 \]
\[ x - 3 = 0 \Rightarrow x = 3 \]

Thus, the values of \( x \) that satisfy the equation are:
\[ x = -2, 2, 3 \]

So, the set A is:
\[ A = \{-2, 2, 3\} \]

The number of elements in set A, denoted as \( m \), is:
\[ m = 3 \]

### Step 2: Determine the set B

The set B is defined as:
\[ B = \{ x \in \mathbb{Z} : -3 < 2x - 1 < 9 \} \]

We can break this compound inequality into two parts:
1. \( -3 < 2x - 1 \)
2. \( 2x - 1 < 9 \)

**Solving the first part:**
\[ -3 < 2x - 1 \]
Adding 1 to both sides:
\[ -2 < 2x \]
Dividing by 2:
\[ -1 < x \]
Thus, we have:
\[ x > -1 \]

**Solving the second part:**
\[ 2x - 1 < 9 \]
Adding 1 to both sides:
\[ 2x < 10 \]
Dividing by 2:
\[ x < 5 \]

Combining both results, we have:
\[ -1 < x < 5 \]

The integers satisfying this inequality are:
\[ x = 0, 1, 2, 3, 4 \]

Thus, the set B is:
\[ B = \{0, 1, 2, 3, 4\} \]

The number of elements in set B, denoted as \( n \), is:
\[ n = 5 \]

### Step 3: Calculate the number of subsets of the Cartesian product \( A \times B \)

The number of subsets of a set is given by \( 2^{\text{number of elements in the set}} \).

The Cartesian product \( A \times B \) will have:
\[ |A \times B| = m \cdot n = 3 \cdot 5 = 15 \]

Thus, the number of subsets of \( A \times B \) is:
\[ 2^{15} \]

### Final Answer
The number of subsets of the set \( A \times B \) is \( 32768 \).

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