Venn Diagrams
1. What It Is
- Venn Diagrams are visual tools used to represent relationships between different sets or groups.
- They help in solving problems involving logic, classification, overlaps, and exclusions.
- Common in reasoning exams to check set theory understanding.
Example:
- 30 students: 15 like cricket, 12 like football, 5 like both.
- Venn Diagram helps visualize overlaps and calculate counts.
2. How to Approach
- Identify sets and total numbers.
- Draw circles for each set.
- Start with intersection (common elements) → most constrained first.
- Fill only one set values next (not overlapping).
- Subtract from total if required → people outside all sets.
- Answer questions using diagram.
3. Rules & Key Points
- Intersection (A ∩ B) → elements common to both sets
- Union (A ∪ B) → elements in either set
- Only A → elements in A but not in B
- Outside all sets → total – (A ∪ B ∪ …)
- For 3 sets → use 3 overlapping circles, fill intersection first, then pairs, then only individual sets, then outside
4. Tricks & Shortcuts
- Start from inside out → intersection first.
- Always subtract overlaps when filling only-one sets.
- For numbers not in any set → total – sum of all sets including intersections.
- For 3 sets, use formula check:
- Total = Only A + Only B + Only C + A∩B + B∩C + A∩C + A∩B∩C + Outside
- Draw clear, labeled circles → reduces mistakes.
5. Types of Questions
- Two-set problems – simple overlap and union.
- Three-set problems – more complex overlaps.
- Exclusive / only-one set → find students only in one group.
- Outside all sets → students not in any category.
- Combined operations → total, union, intersection, complement.
6. Stepwise Solving Strategy
- Identify sets, overlaps, total numbers.
- Draw Venn diagram (2 or 3 sets).
- Fill intersection first.
- Fill only-one set numbers.
- Calculate outside all sets if needed.
- Solve questions using the diagram.
7. Easy Practice Questions
1. Two Sets
Question: In a class of 30 students:
- 15 like cricket, 12 like football, 5 like both.
- How many like only cricket?
Solution:
- Only cricket = 15 – 5 = 10 ✅
- Only football = 12 – 5 = 7
- Neither = 30 – (10 + 7 + 5) = 8
- Answer: 10
2. Only Football
Question: Same data as above, how many like only football?
Solution:
- Only football = 12 – 5 = 7 ✅
3. Neither Cricket nor Football
Solution:
- Total students = 30
- Total in cricket or football = 10 + 7 + 5 = 22
- Neither = 30 – 22 = 8 ✅
4. Three Sets
Question: 50 students:
- 20 like A, 25 like B, 15 like C
- 10 like A&B, 5 like B&C, 3 like A&C, 2 like all three.
- How many like only A?
Solution:
- Only A = 20 – (10 + 3 + 2) = 5 ✅
5. Only C
Solution:
- Only C = 15 – (5 + 3 + 2) = 5 ✅
8. Difficult Practice Questions
6. Exclusive Sets
Question: In 60 students:
- 30 like Maths, 25 like Science, 20 like English
- 10 like Maths & Science, 8 like Science & English, 5 like Maths & English, 3 like all three
- How many like only Science?
Solution:
- Only Science = 25 – (10 + 8 + 3) = 4 ✅
7. Outside All Sets
Solution:
- Total in at least one = 30 + 25 + 20 – (10 + 5 + 8) + 3 = 65 – 23 + 3 = 45
- Outside all = 60 – 45 = 15 ✅
8. Only Two Sets
Question: How many like Maths & English but not Science?
Solution:
- Maths & English only = 5 – (3 in all) = 2 ✅
9. Students in At Least Two Sets
Solution:
- A&B only = 10 – 3 = 7
- B&C only = 8 – 3 = 5
- A&C only = 5 – 3 = 2
- All three = 3
- At least two = 7 + 5 + 2 + 3 = 17 ✅
10. Students in Only One Set
Solution:
- Only Maths = 30 – (10 + 5 + 3) = 12
- Only Science = 25 – (10 + 8 + 3) = 4
- Only English = 20 – (5 + 8 + 3) = 4
- Total only one = 12 + 4 + 4 = 20 ✅
✅ Quick Tips
- Always fill intersections first.
- For 3 sets, subtract overlaps when calculating only-one set.
- Outside all sets = total – sum(all sets with intersections)
- Draw clear diagrams → label everything.
Use formulas to verify sums.