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 problem, we need to find the sets A and B and then calculate the number of subsets of the Cartesian product A × B.

### Step 1: Finding Set A

We start with the equation given for set A:

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

The expression \(2^k = 1\) holds true if \(k = 0\). Therefore, we set the exponent equal to zero:

\[ (x + 2)(x^2 - 5x + 6) = 0 \]

### Step 2: Factoring the Quadratic

Next, we need to factor the quadratic expression \(x^2 - 5x + 6\):

\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]

### Step 3: Setting Up the Equation

Now we can rewrite the equation:

\[ (x + 2)(x - 2)(x - 3) = 0 \]

### Step 4: Finding the Roots

Setting each factor to zero gives us the solutions:

1. \(x + 2 = 0 \Rightarrow x = -2\)
2. \(x - 2 = 0 \Rightarrow x = 2\)
3. \(x - 3 = 0 \Rightarrow x = 3\)

Thus, the set A is:

\[ A = \{-2, 2, 3\} \]

### Step 5: Finding Set B

Now we move on to set B, which is defined as:

\[ B = \{ x \in \mathbb{Z} : -3 < 2x - 1 < 9 \} \]

### Step 6: Solving the Inequality

We can break this into two inequalities:

1. \(2x - 1 > -3\)
2. \(2x - 1 < 9\)

#### Solving the First Inequality

\[ 2x - 1 > -3 \implies 2x > -2 \implies x > -1 \]

#### Solving the Second Inequality

\[ 2x - 1 < 9 \implies 2x < 10 \implies x < 5 \]

### Step 7: Finding the Integer Solutions

Combining these results, we find:

\[ -1 < x < 5 \implies x \in \{0, 1, 2, 3, 4\} \]

Thus, the set B is:

\[ B = \{0, 1, 2, 3, 4\} \]

### Step 8: Calculating the Cartesian Product A × B

The Cartesian product \(A \times B\) consists of all ordered pairs \((a, b)\) where \(a \in A\) and \(b \in B\). The number of elements in \(A\) is 3 and in \(B\) is 5. Therefore, the number of elements in \(A \times B\) is:

\[ |A \times B| = |A| \times |B| = 3 \times 5 = 15 \]

### Step 9: Finding the Number of Subsets

The number of subsets of a set with \(n\) elements is given by \(2^n\). Thus, the number of subsets of \(A \times B\) is:

\[ \text{Number of subsets} = 2^{15} \]

### Final Answer

The number of subsets of the set \(A \times B\) is \(2^{15}\).

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