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

  1. Identify sets and total numbers.
  2. Draw circles for each set.
  3. Start with intersection (common elements) → most constrained first.
  4. Fill only one set values next (not overlapping).
  5. Subtract from total if required → people outside all sets.
  6. 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

  1. Start from inside out → intersection first.
  2. Always subtract overlaps when filling only-one sets.
  3. For numbers not in any set → total – sum of all sets including intersections.
  4. For 3 sets, use formula check:
    • Total = Only A + Only B + Only C + A∩B + B∩C + A∩C + A∩B∩C + Outside
  5. Draw clear, labeled circles → reduces mistakes.

5. Types of Questions

  1. Two-set problems – simple overlap and union.
  2. Three-set problems – more complex overlaps.
  3. Exclusive / only-one set → find students only in one group.
  4. Outside all sets → students not in any category.
  5. Combined operations → total, union, intersection, complement.

6. Stepwise Solving Strategy

  1. Identify sets, overlaps, total numbers.
  2. Draw Venn diagram (2 or 3 sets).
  3. Fill intersection first.
  4. Fill only-one set numbers.
  5. Calculate outside all sets if needed.
  6. 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

  1. Always fill intersections first.
  2. For 3 sets, subtract overlaps when calculating only-one set.
  3. Outside all sets = total – sum(all sets with intersections)
  4. Draw clear diagrams → label everything.

Use formulas to verify sums.