Sets, Relations & Functions
Intersection of sets
Grade 11
Question:
<p>If the sets <em>A</em> and <em>B</em> are defined as<br>\(A = \{(x, y) \mid y = 1/x,\, x \neq 0,\, x \in \mathbb{R}\}\),<br>\(B = \{(x, y) \mid y = -x,\, x \in \mathbb{R}\}\). Then</p>
<p>\(A \cap B = A\)</p>
<p>\(A \cap B = B\)</p>
<p>\(A \cap B = \phi\)</p>
<p>\(A \cup B = A\)</p>
Step-by-Step Solution
Key Concept: Sets A and B are relations (sets of ordered pairs), not functions. A∩B requires finding points that satisfy BOTH y = 1/x AND y = -x simultaneously, which means solving 1/x = -x.
<p><strong>Step 1:</strong> Understand the sets as geometric relations (curves in the plane).</p><p>Set A: y = 1/x is a rectangular hyperbola (two branches) with domain x ≠ 0</p><p>Set B: y = -x is a straight line passing through origin with slope -1</p><p><strong>Step 2:</strong> Find A∩B by solving both equations simultaneously:</p><p>For a point to be in both A and B: y = 1/x AND y = -x</p><p>Therefore: 1/x = -x</p><p>Multiplying by x (where x ≠ 0): 1 = -x²</p><p>This gives: x² = -1</p><p><strong>Step 3:</strong> Since x² = -1 has no real solutions, the hyperbola and line do not intersect in the real plane.</p><p>Therefore: A∩B = ∅ (empty set)</p><p>∴ Answer: C</p>
Correct Answer: C