Sets, Relations & Functions
Intersection of Sets
Grade 11
Question:
<p>If the sets <em>A</em> and <em>B</em> are defined as<br>\(A = \{(x, y) : y = 1/x,\; o \neq x \in \mathbb{R}\}\)<br>\(B = \{(x, y) : y = -x,\; x \in \mathbb{R}\}\),<br>Then \(n(A \cap B) =\) __________.</p>
Step-by-Step Solution
Key Concept: Find intersection of sets A and B by solving the system y = 1/x and y = -x simultaneously. The intersection is empty if no real solutions exist for the system.
<p><strong>Step 1:</strong> Identify set A: y = 1/x where x ≠ 0 (rectangular hyperbola in quadrants I and III)</p><p><strong>Step 2:</strong> Identify set B: y = -x (straight line through origin with slope -1)</p><p><strong>Step 3:</strong> For A ∩ B, we need points (x, y) satisfying both equations simultaneously:</p><p>1/x = -x</p><p><strong>Step 4:</strong> Multiply both sides by x (noting x ≠ 0):</p><p>1 = -x²</p><p>x² = -1</p><p><strong>Step 5:</strong> This equation has no real solutions since x² ≥ 0 for all real x, but we need x² = -1.</p><p><strong>Step 6:</strong> Therefore, no point lies on both curves, so A ∩ B = ∅</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: 0