<p>Let Z be the set of integers. If <span style='display:inline-block;'>\(A = \{x \in \mathbb{Z} : 2^{(x+2)(x^2-5x+6)} = 1\}\)</span> and <span style='display:inline-block;'>\(B = \{x \in \mathbb{Z} : -3 < 2x-1 < 9\}\)</span>, then the number of subsets of the set \(A \times B\) is</p>
Step-by-Step Solution
Key Concept: Find elements of sets A and B separately, then use the formula: number of subsets of a set with n elements is $2^n$. For set $A \times B$, the number of elements is $n(A) \times n(B)$.
<p><strong>Step 1:</strong> Find set A. We need $2^{(x+2)(x^2-5x+6)} = 1 = 2^0$</p><p>$\Rightarrow (x+2)(x^2-5x+6) = 0$</p><p>$\Rightarrow (x+2)(x-3)(x-2) = 0$</p><p>$\Rightarrow x = -2, 2, 3$</p><p>$\therefore A = \{-2, 2, 3\}$</p><p><strong>Step 2:</strong> Find set B. We have $-3 < 2x-1 < 9$ where $x \in \mathbb{Z}$</p><p>$\Rightarrow -2 < 2x < 10$</p><p>$\Rightarrow -1 < x < 5$</p><p>$\therefore B = \{0, 1, 2, 3, 4\}$</p><p><strong>Step 3:</strong> Find $n(A \times B)$. We have $n(A) = 3$ and $n(B) = 5$</p><p>$\therefore n(A \times B) = 3 \times 5 = 15$</p><p><strong>Step 4:</strong> Number of subsets of $A \times B = 2^{15}$</p><p>∴ Answer is (c).</p>
Correct Answer: c