Venn Diagrams
Key Concepts & Formulas
| # | Concept | Quick Explanation |
|---|---|---|
| 1 | Union (A∪B) | Total elements in A OR B = n(A) + n(B) - n(A∩B) |
| 2 | Intersection (A∩B) | Elements common to both sets A and B |
| 3 | Only A | Elements in A but not in B = n(A) - n(A∩B) |
| 3 | Only B | Elements in B but not in A = n(B) - n(A∩B) |
| 5 | Neither A nor B | Total - n(A∪B) |
| 6 | Three Sets Formula | n(A∪B∪C) = n(A)+n(B)+n(C)-n(A∩B)-n(B∩C)-n(C∩A)+n(A∩B∩C) |
| 7 | Percentage Method | Convert all values to percentages for easier calculation |
10 Practice MCQs
Q1. In a survey of 100 passengers at New Delhi station, 60 use IRCTC app and 40 use UTS app. If 20 passengers use both apps, how many use only IRCTC app? A) 20 B) 40 C) 60 D) 80
Answer: B) 40
Solution: Only IRCTC = Total IRCTC - Both = 60 - 20 = 40
Shortcut: Only A = n(A) - n(A∩B)
Concept: Venn Diagrams - Finding “Only A” region
Q2. At Mumbai Central station, 150 passengers were surveyed. 90 like tea, 70 like coffee, and 40 like both. How many like neither? A) 30 B) 40 C) 50 D) 60
Answer: A) 30
Solution: n(T∪C) = 90 + 70 - 40 = 120 Neither = 150 - 120 = 30
Shortcut: Neither = Total - [n(A)+n(B)-n(A∩B)]
Concept: Venn Diagrams - Finding “Neither” region
Q3. In a train compartment of 80 passengers, 50 can speak Hindi and 35 can speak English. If everyone speaks at least one language, how many speak both? A) 5 B) 15 C) 25 D) 35
Answer: A) 5
Solution: n(H∩E) = n(H) + n(E) - Total = 50 + 35 - 80 = 5
Shortcut: Both = n(A)+n(B)-Total (when Total = n(A∪B))
Concept: Venn Diagrams - Finding intersection when total equals union
Q4. At Howrah station, out of 200 passengers, 120 have reserved tickets, 100 have platform tickets, and 60 have both. How many have only platform tickets? A) 40 B) 60 C) 80 D) 100
Answer: A) 40
Solution: Only Platform = Platform total - Both = 100 - 60 = 40
Shortcut: Visualize: Platform circle minus overlap region
Concept: Venn Diagrams - Finding exclusive regions
Q5. In Rajdhani Express, 180 passengers were surveyed about food preferences. 110 prefer vegetarian, 95 prefer non-vegetarian, and 45 prefer both. How many prefer only vegetarian? A) 45 B) 65 C) 85 D) 105
Answer: B) 65
Solution: Only Vegetarian = Total vegetarian - Both = 110 - 45 = 65
Shortcut: Only A = n(A) - n(A∩B)
Concept: Venn Diagrams - Calculating exclusive preferences
Q6. At Chennai Central, three types of passes are available: Monthly (M), Quarterly (Q), and Yearly (Y). In a group of 250 regular commuters, 150 have M, 120 have Q, 100 have Y. 80 have M∩Q, 70 have Q∩Y, 60 have M∩Y, and 50 have all three. How many have at least one pass? A) 200 B) 220 C) 240 D) 250
Answer: B) 220
Solution: Using 3-set formula: n(M∪Q∪Y) = 150+120+100-80-70-60+50 = 220
Shortcut: Remember: Add singles, subtract pairs, add triple intersection
Concept: Venn Diagrams - Three overlapping sets
Q7. In Duronto Express with 300 passengers, surveys show: 180 use AC class, 160 use Sleeper class, 140 use Chair car. 100 use AC&Sleeper, 80 use Sleeper&Chair, 70 use AC&Chair, 50 use all three. How many use only AC class? A) 60 B) 70 C) 80 D) 90
Answer: A) 60
Solution: Only AC = AC total - (AC∩Sleeper + AC∩Chair - All three) = 180 - (100 + 70 - 50) = 180 - 120 = 60
Shortcut: Only A = n(A) - [n(A∩B)+n(A∩C)-n(A∩B∩C)]
Concept: Venn Diagrams - Finding exclusive region in 3 sets
Q8. At a junction station, trains arrive from 3 directions. Total 24 trains: 15 from North, 18 from South, 16 from East. 10 come from North&South, 8 from South&East, 9 from North&East. If 5 trains come from all three directions, how many come from exactly two directions? A) 12 B) 15 C) 17 D) 20
Answer: A) 12
Solution: Exactly two directions = (N∩S only) + (S∩E only) + (N∩E only) = (10-5) + (8-5) + (9-5) = 5 + 3 + 4 = 12
Shortcut: Exactly two = Sum of pairwise intersections - 3×(all three)
Concept: Venn Diagrams - Calculating “exactly two” regions
Q9. In a survey of 400 railway employees about language skills: 250 know English, 200 know Hindi, 180 know regional language. 150 know E&H, 120 know H&R, 100 know E&R, 80 know all three. How many know exactly one language? A) 120 B) 140 C) 160 D) 180
Answer: C) 160
Solution: Only English = 250 - (150+100-80) = 80 Only Hindi = 200 - (150+120-80) = 10 Only Regional = 180 - (120+100-80) = 40 Total = 80 + 10 + 40 = 130
Shortcut: Calculate each “only” separately then sum
Concept: Venn Diagrams - Finding “exactly one” in complex 3-set problems
Q10. At a metro station, passenger categories are: Senior citizens (S), Students (T), Disabled (D). Total passengers: 500. Given: n(S)=200, n(T)=180, n(D)=150, n(S∩T)=80, n(T∩D)=70, n(S∩D)=60, n(S∩T∩D)=40. How many are none of these categories? A) 60 B) 80 C) 100 D) 120
Answer: A) 60
Solution: n(S∪T∪D) = 200+180+150-80-70-60+40 = 360 None = 500 - 360 = 140
Shortcut: Use complement: None = Total - n(A∪B∪C)
Concept: Venn Diagrams - Advanced 3-set with complement calculation
5 Previous Year Questions
PYQ 1. In a class of 50 students, 30 like cricket and 25 like football. If 10 students like both games, how many like neither? [RRB NTPC 2021 CBT-1]
Answer: B) 5
Solution: n(C∪F) = 30 + 25 - 10 = 45 Neither = 50 - 45 = 5
Exam Tip: Always verify if total equals union when “neither” is asked
PYQ 2. At a railway counter, 120 people are in queue. 80 want reservation tickets, 70 want platform tickets, and 50 want both. How many want only platform tickets? [RRB Group D 2022]
Answer: A) 20
Solution: Only Platform = 70 - 50 = 20
Exam Tip: In railway context, “both” usually means having two types of tickets
PYQ 3. Survey of 200 passengers: 120 vegetarian, 100 non-vegetarian, 60 both. Find only vegetarian passengers. [RRB ALP 2018]
Answer: C) 60
Solution: Only Vegetarian = 120 - 60 = 60
Exam Tip: Food preference questions are common in railway exams
PYQ 4. Three train routes A, B, C. Total 15 trains daily. 8 on route A, 10 on B, 7 on C. 4 on A&B, 3 on B&C, 2 on A&C, 1 on all three. Find trains on exactly one route. [RRB JE 2019]
Answer: B) 7
Solution: Only A = 8 - (4+2-1) = 3 Only B = 10 - (4+3-1) = 4 Only C = 7 - (3+2-1) = 3 Total only one = 3 + 4 + 3 = 10
Exam Tip: Route-based questions test 3-set Venn diagrams
PYQ 5. In RPF recruitment, 300 applicants. 180 have height criteria, 200 have education criteria, 150 have both. How many have only education criteria? [RPF SI 2019]
Answer: A) 50
Solution: Only Education = 200 - 150 = 50
Exam Tip: Recruitment criteria questions frequently use Venn diagrams
Speed Tricks & Shortcuts
| Situation | Shortcut | Example |
|---|---|---|
| Finding “Both” when Total=Union | Both = A + B - Total | If 100 people, 60 like tea, 50 like coffee: Both = 60+50-100 = 10 |
| “Only A” calculation | Only A = A - Both | If 80 have cars, 30 have both car & bike: Only car = 80-30 = 50 |
| Percentage method | Convert all to % first | In 200 people, 40% like X, 30% like Y, 10% like both: Only X = 40-10 = 30% |
| Three sets “exactly two” | Sum pairs - 3×triple | If A∩B=20, B∩C=15, A∩C=10, all three=5: Exactly two = (20+15+10)-(3×5) = 30 |
| Neither calculation | Neither = Total - (A+B-Both) | Total 150, A=90, B=70, Both=40: Neither = 150-(90+70-40) = 30 |
Common Mistakes to Avoid
| Mistake | Why Students Make It | Correct Approach |
|---|---|---|
| Confusing “Only A” with “A total” | Not reading “only” carefully | Always subtract intersection from total |
| Forgetting to subtract intersection twice | Adding A+B without subtracting overlap | Remember: A∪B = A + B - A∩B |
| Assuming Total = Union | Not checking if “neither” exists | Verify if all elements are accounted for |
| Miscounting three-set regions | Complex overlapping confusion | Draw diagram and label each region clearly |
| Calculation errors in percentage method | Rushing conversion steps | Always verify: Only A% = A% - Both% |
Quick Revision Flashcards
| Front (Question/Term) | Back (Answer) |
|---|---|
| Union formula (2 sets) | n(A∪B) = n(A) + n(B) - n(A∩B) |
| Intersection meaning | Elements common to both sets |
| “Only A” formula | n(A) - n(A∩B) |
| Neither formula | Total - n(A∪B) |
| Three sets union | A+B+C-(A∩B+B∩C+C∩A)+A∩B∩C |
| Exactly two regions | Sum of pairwise - 3×triple intersection |
| Percentage shortcut | Convert to % for easier mental math |
| Visual check method | Draw circles and verify regions add up |
| Railway context tip | Tickets, routes, passenger categories common |
| Time saver | Use addition-subtraction method over equations |
Topic Connections
Direct Link:
- Set Theory: Venn diagrams visually represent set operations
- Percentage Problems: Often combined with percentage calculations
- Data Interpretation: Foundation for interpreting overlapping data
Combined Questions:
- Venn + Percentage: “40% of passengers like both…”
- Venn + Ratio: “Ratio of AC to Non-AC passengers who also have…”
- Venn + Average: “Average age of passengers in overlapping categories”
Foundation For:
- Logical Reasoning: Advanced syllogism problems
- Data Sufficiency: Determining if Venn data is complete
- Complex DI: Multi-dimensional data representation
BETA
