Sets, Relations & Functions
Union and Intersection of Sets
Grade None
<p>Sets <em>A</em> and <em>B</em> have 5 and 7 elements respectively. What can be the minimum number of elements in \(A \cup B\) is __________.</p>
Step-by-Step Solution
Key Concept: The union A ∪ B is minimized when the overlap (intersection) is maximized. Since A has fewer elements, the maximum possible intersection is |A| = 5, making |A ∪ B| = |A| + |B| - |A ∩ B| = 5 + 7 - 5 = 7.
<p><strong>Step 1:</strong> Recall the fundamental formula for union of two sets:</p><p>|A ∪ B| = |A| + |B| - |A ∩ B|</p><p><strong>Step 2:</strong> We have |A| = 5 and |B| = 7. To minimize |A ∪ B|, we need to maximize |A ∩ B|.</p><p><strong>Step 3:</strong> The maximum value of |A ∩ B| is limited by the smaller set: |A ∩ B| ≤ min(|A|, |B|) = min(5, 7) = 5</p><p><strong>Step 4:</strong> This maximum is achievable when A ⊆ B (all elements of A are contained in B).</p><p><strong>Step 5:</strong> Substituting into the formula:</p><p>|A ∪ B|<sub>min</sub> = 5 + 7 - 5 = 7</p><p>∴ Answer: <strong>7</strong></p>
Correct Answer: 7