24  Venn Diagrams

A Venn diagram, introduced by John Venn in 1880, uses overlapping circles to represent sets and the logical relations between them. Each circle represents a set; overlaps represent intersections.

24.1 Set-Theory Vocabulary

TipWorking Set Vocabulary
Symbol Meaning
A ∪ B Union — elements in A or B (or both)
A ∩ B Intersection — elements in both A and B
A − B or A  B Difference — elements in A but not in B
A′ or Ā Complement — elements not in A
A ⊆ B A is a subset of B (every member of A is in B)
A ⊂ B A is a proper subset of B (subset and not equal)
∅ Empty set (no elements)
U Universal set — all elements under consideration

24.2 Two-Set Venn Diagrams — Three Possibilities

TipThree Possible Relations between Two Sets
Relation Description Example
Disjoint (no overlap) A and B share no element Cats and dogs
Overlapping (partial overlap) A and B share some elements but neither contains the other Doctors and parents
Nested (one inside the other) A is fully inside B (or vice versa) Cats are within mammals 

flowchart LR
  subgraph Disjoint
    A1((A))
    B1((B))
  end
  subgraph Overlapping
    A2((A)) --- B2((B))
  end
  subgraph Nested
    B3((B<br/>A))
  end
    classDef default fill:#003366,color:#ffffff,stroke:#ffcc00,stroke-width:3px,rx:10px,ry:10px;

24.3 Categorical Statements as Venn Diagrams

The four standard categorical statements (A, E, I, O) can be drawn as Venn diagrams.

TipVenn Representation of A, E, I, O
Form Statement Venn picture
A All S are P S circle inside P circle (S region outside P is empty)
E No S are P S and P circles disjoint (no overlap)
I Some S are P S and P circles overlap with at least one element in the overlap
O Some S are not P At least one element in S but outside P

24.4 Three-Set Venn Diagrams

A three-set Venn diagram has eight distinct regions, including the area outside all three circles.

TipEight Regions of Three Overlapping Circles A, B, C
Region Members
Only A In A, not in B, not in C
Only B In B, not in A, not in C
Only C In C, not in A, not in B
A ∩ B only In A and B but not C
A ∩ C only In A and C but not B
B ∩ C only In B and C but not A
A ∩ B ∩ C In all three
Outside all In none

24.5 Counting with Venn Diagrams — the Inclusion-Exclusion Principle

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|

TipWorked example — two sets

In a class of 50 students, 30 study mathematics and 25 study physics. 15 study both. How many study at least one subject?

|M ∪ P| = 30 + 25 − 15 = 40 students.

TipWorked example — three sets

In a survey of 100 people: 60 read newspaper A, 50 read B, 40 read C. 30 read both A and B, 20 read both B and C, 25 read both A and C. 10 read all three. How many read at least one?

|A ∪ B ∪ C| = 60 + 50 + 40 − 30 − 20 − 25 + 10 = 85 people.

24.6 Using Venn Diagrams to Test Syllogisms

Venn diagrams give a graphical test for syllogism validity. Draw circles for the three terms; shade regions known to be empty; place an “x” where at least one element is known to exist; the conclusion is valid if the diagram of premises automatically produces the conclusion.

TipWorked example — testing a syllogism

Premise 1: All cats are mammals. (A: All S are P) Premise 2: All mammals are animals. (A: All P are Q) Conclusion: Therefore, all cats are animals.

Three circles — Cats (C), Mammals (M), Animals (A). Premise 1 places C inside M; premise 2 places M inside A. By transitivity, C is inside A → conclusion follows. Valid.

24.7 Common Question Types

TipThree NTA Question Patterns
  1. Which diagram represents the given relation? (e.g., Doctors / Females / Indians)
  2. Identify the syllogism’s validity by Venn diagram.
  3. Counting / overlap problems using inclusion-exclusion.
TipWorked example — relation identification

Doctors, Females, Indians.

These three sets overlap: an Indian doctor may be female; a female may be Indian and not a doctor; a doctor may be Indian but male. None is contained in another. → All three circles overlap.

If the question gives “Doctors, Surgeons, Indians”, the answer is different — surgeons are a subset of doctors, but Indians overlap with both.

24.8 Practice Questions

Q 01 Set Notation Easy

In set notation, A ∩ B represents:

  • AElements in A or B
  • BElements in both A and B
  • CElements in A but not in B
  • DAll elements outside A and B
View solution
Correct Option: B
A ∩ B = intersection — elements in both A and B.
Q 02 Inclusion-Exclusion Medium

In a class of 60 students, 40 study English and 35 study Hindi. 20 study both. How many study at least one of the two?

  • A45
  • B55
  • C75
  • D95
View solution
Correct Option: B
|E ∪ H| = |E| + |H| − |E ∩ H| = 40 + 35 − 20 = 55.
Q 03 Three-Set Relation Medium

Which Venn diagram best represents the relation among "Doctors, Females, Indians"?

  • AThree disjoint circles
  • BAll three circles overlapping
  • CDoctors inside Indians inside Females
  • DFemales inside Doctors
View solution
Correct Option: B
All three sets partially overlap — none is fully contained in another, and pairs overlap.
Q 04 Categorical Venn Medium

"All mammals are animals" — the Venn diagram representation places:

  • AAnimals circle inside Mammals circle
  • BMammals circle inside Animals circle
  • CThe two circles disjoint
  • DThe two circles partially overlapping
View solution
Correct Option: B
"All S are P" places S inside P. So Mammals (S) inside Animals (P).
Q 05 Three-Set Inclusion-Exclusion Hard

In a survey of 100 people: 60 read paper A, 50 read B, 40 read C. 30 read A and B, 20 read B and C, 25 read A and C. 10 read all three. How many read at least one paper?

  • A75
  • B85
  • C95
  • D100
View solution
Correct Option: B
|A ∪ B ∪ C| = 60 + 50 + 40 − 30 − 20 − 25 + 10 = 85.
Q 06 Set Operations Easy

If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then A ∪ B is:

  • A{3, 4}
  • B{1, 2, 3, 4, 5, 6}
  • C{1, 2}
  • D{5, 6}
View solution
Correct Option: B
Union = all distinct elements from both sets: {1, 2, 3, 4, 5, 6}.
Q 07 Disjoint Sets Easy

Two sets that share no element are called:

  • ASubset
  • BUniversal
  • CDisjoint
  • DPower
View solution
Correct Option: C
Disjoint sets have empty intersection. In a Venn diagram their circles do not overlap.
Q 08 Subset Reasoning Medium

Given that "All cats are mammals" and "All mammals are animals", what can be deduced using a Venn diagram?

  • AAll animals are cats
  • BAll cats are animals
  • CNo cats are animals
  • DSome cats are not mammals
View solution
Correct Option: B
By transitivity (Cats ⊂ Mammals ⊂ Animals), All cats are animals.
ImportantQuick recall
  • Two-set inclusion-exclusion: |A ∪ B| = |A| + |B| − |A ∩ B|.
  • Three-set: |A ∪ B ∪ C| = ΣA − Σ(pairwise intersections) + |A ∩ B ∩ C|.
  • Two-set relations: Disjoint, Overlapping, Nested.
  • Categorical statements as Venns: A → S inside P; E → disjoint; I → overlap with x; O → x in S outside P.
  • Three-set Venn has 8 distinct regions (including outside).