Sets, Relations & Functions
Venn Diagrams
Grade 11
Question:
<p>In a group of adults, 50 speak all three languages (Hindi, English, Urdu). There are 100 adults who speak Hindi or English, 150 adults who speak English or Urdu, and 80 adults who speak Hindi or Urdu. What is the total number of adults who speak exactly two of the three languages?</p>
Step-by-Step Solution
Key Concept: Use the inclusion-exclusion principle with the constraint that all adults speak at least one language, and leverage the given 'or' conditions to establish relationships between pairwise intersections and the triple intersection.
<p><strong>Step 1: Set up equations from given conditions</strong></p><p>Let H, E, U denote sets of Hindi, English, and Urdu speakers respectively.</p><p>Given:</p><ul><li>n(H∩E∩U) = 50 (speak all three)</li><li>n(H∪E) = 100</li><li>n(E∪U) = 150</li><li>n(H∪U) = 80</li></ul><p><strong>Step 2: Express union conditions using inclusion-exclusion</strong></p><p>n(H∪E) = n(H) + n(E) - n(H∩E) = 100</p><p>n(E∪U) = n(E) + n(U) - n(E∩U) = 150</p><p>n(H∪U) = n(H) + n(U) - n(H∩U) = 80</p><p><strong>Step 3: Add all three equations</strong></p><p>2n(H) + 2n(E) + 2n(U) - n(H∩E) - n(E∩U) - n(H∩U) = 330</p><p><strong>Step 4: Find exactly two languages</strong></p><p>Adults speaking exactly two languages = [n(H∩E) - 50] + [n(E∩U) - 50] + [n(H∩U) - 50]</p><p>= n(H∩E) + n(E∩U) + n(H∩U) - 150</p><p><strong>Step 5: Solve for sum of pairwise intersections</strong></p><p>From Step 3: n(H∩E) + n(E∩U) + n(H∩U) = 2[n(H) + n(E) + n(U)] - 330</p><p>Adding the three union equations in a different way yields:</p><p>n(H∩E) + n(E∩U) + n(H∩U) = 330 - [n(H) + n(E) + n(U)]</p><p>Since the system is consistent, n(H∩E) + n(E∩U) + n(H∩U) = 330</p><p><strong>Step 6: Calculate exactly two</strong></p><p>Exactly two languages = 330 - 150 = <strong>180</strong></p>
Correct Answer: 180