Question

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

• A. $2^{18}$
• B. $2^{10}$
• C. $2^{15}$
• D. $2^{12}$

Answer 1

The Correct Option is C

1.  Determine Set A: We have the condition \(2^{(x+2)(x^2 - 5x + 6)} = 1\). 
   The exponential function \(2^y = 1\) if and only if \(y = 0\). Hence, we need: \((x+2)(x^2 - 5x + 6) = 0\). This means:
   • \(x+2=0\), which gives \(x = -2\).
   • \(x^2 - 5x + 6 = 0\). This is a quadratic equation which factors as: \((x-2)(x-3) = 0\)
     Thus, \(x = 2\) or \(x = 3\).
2. Determine Set B: We solve the inequality \(-3 < 2x - 1 < 9\):
   • First inequality: \(-3 < 2x - 1\) results in \(2x > -2\) or \(x > -1\).
   • Second inequality: \(2x - 1 < 9\) results in \(2x < 10\) or \(x < 5\).
3. Calculate the Cartesian Product \(A \times B\): The number of elements in set \(A\) is 3, and the number of elements in set \(B\) is 5. Thus, the number of elements in \(A \times B\) is \(3 \times 5 = 15\).
4. Calculate the Number of Subsets of \(A \times B\): A set with \(n\) elements has \(2^n\) subsets. Therefore, the number of subsets of \(A \times B\) is \(2^{15}\).
5. Conclusion: Thus, the number of subsets of the set \(A \times B\) is \(2^{15}\).