Sets, Relations & Functions
Union and Intersection of Sets
Grade None
Question:
<p>Here, <em>n</em>(<em>A</em>) = 5, <em>n</em>(<em>B</em>) = 7. What is the minimum number of elements in <em>A</em> ∪ <em>B</em>?</p>
Step-by-Step Solution
Key Concept: The union A ∪ B is minimized when the overlap (intersection) is maximized. Since n(A) = 5, the maximum possible intersection is min(5, 7) = 5, meaning all of A can be contained in B.
<p><strong>Step 1:</strong> Use the fundamental principle: n(A ∪ B) = n(A) + n(B) - n(A ∩ B)</p><p><strong>Step 2:</strong> To minimize n(A ∪ B), we must maximize n(A ∩ B).</p><p><strong>Step 3:</strong> The maximum value of n(A ∩ B) is min(n(A), n(B)) = min(5, 7) = 5. This occurs when A ⊆ B.</p><p><strong>Step 4:</strong> Therefore, n(A ∪ B)_min = 5 + 7 - 5 = 7</p><p>∴ Answer: <strong>7</strong></p>
Correct Answer: 7