Sets, Relations & Functions
Practical Problems on Union and Intersection
Grade 11
Question:
<p>Let <em>X</em>, <em>Y</em> be two sets and <em>X</em> has 40 elements, \(X \cup Y\) has 60 elements and \(X \cap Y\) has 10 elements. Then the number of elements in <em>Y</em> is</p>
<p>30</p>
<p>10</p>
<p>40</p>
<p>50</p>
Step-by-Step Solution
Key Concept: Use the fundamental set formula |X ∪ Y| = |X| + |Y| - |X ∩ Y| to find the unknown cardinality of set Y by algebraic substitution.
<p><strong>Step 1:</strong> Apply the inclusion-exclusion principle for sets:</p><p>|X ∪ Y| = |X| + |Y| - |X ∩ Y|</p><p><strong>Step 2:</strong> Substitute the given values: |X| = 40, |X ∪ Y| = 60, |X ∩ Y| = 10</p><p>60 = 40 + |Y| - 10</p><p><strong>Step 3:</strong> Solve for |Y|:</p><p>60 = 30 + |Y|</p><p>|Y| = 30</p><p>∴ Answer: <strong>30 elements in Y</strong></p>
Correct Answer: A