6.4 Venn Diagram
Key Takeaways
- Two-set Venn has 4 regions; three-set Venn has 8 regions — fill inside out, starting with the all-three overlap.
- For two sets: Total = A + B − A∩B + neither; subtract the overlap once because it was counted twice.
- For three sets: Total = A + B + C − AB − AC − BC + ABC + none; the triple overlap was subtracted three times so add it back once.
- 'Exactly two' excludes the all-three region — sum (AB only) + (AC only) + (BC only); 'at least two' includes it.
- Logical-deduction Venn asks which diagram fits a relationship — Fathers ⊂ Men, Doctors overlaps both and extends outside for women.
Why This Matters
Venn diagram questions on the RRB Group D CBT split into two flavors: set problems that ask you to count members of unions and intersections, and logical-deduction questions that ask you to pick the diagram fitting a relationship. The set-problem variety appears in 3 to 5 marks per cycle and is highly mechanical — the inside-out fill method clears almost every item.
Set Operations Refresher
Three operations underpin the formulas:
| Operation | Symbol | Meaning |
|---|---|---|
| Union | A ∪ B | All elements in A, or B, or both |
| Intersection | A ∩ B | Elements in both A and B |
| Difference | A − B | Elements in A but not in B |
| Complement | A' | Elements not in A (within the universal set) |
For two sets: |A ∪ B| = |A| + |B| − |A ∩ B|. For three sets: |A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |A ∩ C| − |B ∩ C| + |A ∩ B ∩ C|. The pattern alternates signs as you add more sets.
Two-Set Venn
Two overlapping circles create four regions: A only, B only, A∩B (both), and neither. The total is:
Total = A + B − A∩B + neither
Subtract A∩B once because the overlap was counted in both A and B. Conversely, to find neither, compute (A + B − A∩B) and subtract from the universal set.
| Region | Count |
|---|---|
| A only | A − A∩B |
| B only | B − A∩B |
| Both | A∩B |
| At least one (union) | A + B − A∩B |
| Neither | Total − (A + B − A∩B) |
Three-Set Venn
Three overlapping circles create eight regions. The key formula:
Total = A + B + C − AB − AC − BC + ABC + none
Where AB, AC, BC are the pairwise intersections (including the triple overlap). The triple overlap ABC was subtracted three times (once in each pair), so it is added back once.
To fill a three-set Venn from given data, work inside out:
- Fill ABC (all three) first — this is usually given or derivable.
- Fill each pairwise-only region: AB only = AB − ABC, and similarly for AC and BC.
- Fill each single-only region: A only = A − (AB only + AC only + ABC), and similarly for B and C.
- Sum all eight regions; subtract from Total to get none.
Worked Example 1 (Two Sets)
In a group of 70 people, 40 drink tea, 30 drink coffee, and 15 drink both. How many drink neither?
- Tea only = 40 − 15 = 25.
- Coffee only = 30 − 15 = 15.
- Both = 15.
- At least one = 25 + 15 + 15 = 55.
- Neither = 70 − 55 = 15.
Answer: 15.
Worked Example 2 (Three Sets)
In a class of 100 students: 50 take Math, 40 take Physics, 30 take Chemistry; 15 take Math and Physics, 10 take Physics and Chemistry, 12 take Math and Chemistry; 5 take all three. How many take none?
- Math only = 50 − ((15 − 5) + (12 − 5) + 5) = 50 − (10 + 7 + 5) = 28.
- Physics only = 40 − ((15 − 5) + (10 − 5) + 5) = 40 − (10 + 5 + 5) = 20.
- Chemistry only = 30 − ((12 − 5) + (10 − 5) + 5) = 30 − (7 + 5 + 5) = 13.
- Math ∩ Physics only = 10.
- Math ∩ Chemistry only = 7.
- Physics ∩ Chemistry only = 5.
- All three = 5.
- At least one = 28 + 20 + 13 + 10 + 7 + 5 + 5 = 88.
- None = 100 − 88 = 12.
Answer: 12.
"Exactly Two" vs "At Least Two"
Two frequently confused quantities:
| Question | What to sum |
|---|---|
| Exactly one subject | A only + B only + C only |
| Exactly two subjects | (AB only) + (AC only) + (BC only) — excludes the triple overlap |
| At least two subjects | (AB only) + (AC only) + (BC only) + ABC |
| At least one subject | Total − None |
A common RRB trap is to count "exactly two" by summing (AB + AC + BC) raw, which double-counts the triple region. Always subtract ABC from each pairwise before summing.
Worked Example 3 (Exactly Two)
Using the class above, how many students take exactly two subjects?
- Math ∩ Physics only = 10.
- Math ∩ Chemistry only = 7.
- Physics ∩ Chemistry only = 5.
- Sum = 22.
Answer: 22.
Logical Deduction Venn
The second RRB Venn flavor asks which diagram correctly represents a relationship among three classes. The trick is to identify which class is a subset of another and which classes overlap without containment. For "Vegetables, Carrots, Potatoes," Vegetables is the largest circle; Carrots and Potatoes are disjoint (a carrot is not a potato) circles inside it. For "Women, Mothers, Doctors," Mothers ⊂ Women, and Doctors overlaps both but extends outside (for male doctors). Drawing the inclusion and overlap on scratch paper before looking at the options saves time.
Worked Example 4 (Diagram Selection)
"Which diagram correctly represents the relationship: Men, Fathers, Doctors?"
- All fathers are men → Fathers ⊂ Men.
- Some doctors are men, some doctors are fathers, some doctors are women.
- So: a big Men circle; a small Fathers circle inside; a Doctors circle that overlaps Men and Fathers and extends outside (for women doctors).
Exam Traps
- "At least one of the two" means union; "both" means intersection; "exactly one" means union minus intersection.
- "Neither" requires knowing the universal set size — check that the question gives it.
- For diagram-selection questions, check whether the smaller set can extend outside the larger (e.g., Doctors vs Men — yes, because women doctors exist).
- Some questions give percentages, not counts. Convert to counts or work in percentages; do not mix the two.
In a survey of 60 people, 35 like tea, 25 like coffee, and 10 like both. How many like neither?
In a class, 25 play cricket, 20 play football, 15 play hockey, 10 play cricket and football, 8 play football and hockey, 6 play cricket and hockey, and 4 play all three. How many play exactly two games?
Which diagram best represents the relationship: Women, Mothers, Doctors?