Sets, Relations & Functions
Power Set
Grade 11
Question:
<p>Set \(A = \{x : |x| < 3, x \in \mathbb{Z}\}\) and relation \(R = \{(x, y) : y = |x|, x \neq -1\}\). The number of elements in the power set of <em>R</em> is</p>
<p>8</p>
<p>16</p>
<p>32</p>
<p>64</p>
Step-by-Step Solution
Key Concept: The composition f∘g is defined only where g's range intersects f's domain. You must find g(A) first, then apply f to those values, carefully tracking domain restrictions at each step.
<p><strong>Step 1:</strong> Find set A: A = {x : |x| < 3} = (-3, 3)</p><p><strong>Step 2:</strong> Compute g(A) where g: ℝ → ℝ. Determine the range of g on the interval (-3, 3). This gives the intermediate set that g maps A into.</p><p><strong>Step 3:</strong> Verify g(A) ⊆ Domain(f). The composition f∘g is only defined on elements of A where g(A) intersects Domain(f).</p><p><strong>Step 4:</strong> Apply f to g(A) to find (f∘g)(A). This requires computing f on the specific range values obtained from Step 2.</p><p><strong>Step 5:</strong> Simplify the result using properties of f and the restricted domain to arrive at the interval or set form matching option B.</p><p>∴ Answer: B</p>
Correct Answer: B