25 Venn diagram: Simple and multiple use for establishing validity of arguments

25.1 What the Syllabus Covers

A Venn diagram, introduced by John Venn (1880), uses overlapping circles to represent sets and the logical relations between them. Each circle represents a set; overlaps represent intersections. The syllabus expects the candidate to use Venn diagrams (a) to show the logical relations among classes (categorical propositions A, E, I, O) and (b) to establish the validity of categorical syllogisms.

PYQs reliably test: (a) drawing the right Venn picture for a given proposition, (b) matching word descriptions (“some teachers are not doctors”) to the correct diagram, (c) using a 3-circle Venn to check syllogism validity, and (d) counting elements in unions and intersections (set-theory questions).

25.2 Set-Theory Vocabulary

Working 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)	Elements in A but not in B
A’ (or A^c)	Complement — elements not in A
A ⊆ B	A is a subset of B
A ⊂ B	A is a proper subset of B
A ≡ B	A and B are identical
∅	Empty set
U	Universal set
n(A)	Number of elements (cardinality) of A

25.2.1 The Key Cardinality Identity

Two- and Three-Set Inclusion-Exclusion

Two sets: n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
Three sets: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A∩B) − n(B∩C) − n(C∩A) + n(A∩B∩C).
Complement: n(A’) = n(U) − n(A).
De Morgan’s Laws: (A ∪ B)’ = A’ ∩ B’; (A ∩ B)’ = A’ ∪ B’.

25.3 The Four Standard Venn Pictures (A, E, I, O)

A categorical proposition (Topic 21) has a unique Venn picture using two circles S and P.

Four Categorical Propositions and Their Venn Pictures

A — All S are P — S is entirely inside P. (Or: shade the part of S that is outside P, since it must be empty.)
E — No S are P — S and P do not overlap. (Or: shade the S∩P region as empty.)
I — Some S are P — an x placed inside S∩P (at least one element exists there).
O — Some S are not P — an x placed inside S but outside P.

flowchart TB
  A["A: All S are P<br/>(S ⊆ P)"] --> AD[Picture: S inside P]
  E["E: No S are P<br/>(S ∩ P = ∅)"] --> ED[Picture: S, P disjoint]
  I["I: Some S are P<br/>(S ∩ P ≠ ∅)"] --> ID[Picture: x in S ∩ P]
  O["O: Some S are not P<br/>(S − P ≠ ∅)"] --> OD[Picture: x in S − P]
    classDef default fill:#003366,color:#ffffff,stroke:#ffcc00,stroke-width:3px,rx:10px,ry:10px;

25.3.1 Three Standard Set-Relations Between Two Circles

Three Standard 2-Circle Relations

Inclusion — one circle entirely inside another (subset).
Disjoint — circles do not overlap (mutual exclusion).
Overlap — circles partially intersect (some in common, some not).
Equality — two circles coincide entirely (identical sets).

25.4 Reading PYQ-Style Word Problems

NTA loves questions like:

“Choose the diagram that best illustrates the relation among: Teachers, Doctors, Women.”

The candidate must judge whether each pair is disjoint, overlapping, or one-inside-another.

A Reading Heuristic

Some X are Y → overlapping circles.
All X are Y → X inside Y.
No X are Y → disjoint circles.
All X are Y AND all Y are X → X and Y coincide.

25.4.1 Worked Examples

Word Problem — Worked

Q. Pick the diagram for: Teachers, Doctors, Women. Reasoning: Some teachers are women; some teachers are not women. Some doctors are women; some are not. Some teachers are doctors (rare but possible). So all three circles overlap.

Q. Pick the diagram for: Birds, Sparrows, Cats. Reasoning: All sparrows are birds; no cats are birds. Picture: Sparrows entirely inside Birds; Cats disjoint from Birds.

Q. Pick the diagram for: Boys, Children, Students. Reasoning: All boys are children; some students are children (and some are not, e.g., adult students). Some boys are students. → All boys inside Children; Students overlapping with Children, partially overlapping with Boys.

25.5 Venn Diagrams for Syllogism Validity

A categorical syllogism has three terms (S, P, M). A 3-circle Venn diagram is the standard tool for checking validity.

25.5.1 Procedure

Procedure to Test a Syllogism by Venn Diagram

Draw three overlapping circles S, P, M.
Diagram each premise in turn (universal first, then particular).
- Universal (A or E) — shade the empty region.
- Particular (I or O) — place an x in the appropriate region. If the x could go in two sub-regions, place it on the boundary between them.
Read the conclusion from the diagram.
If the conclusion is forced by what you’ve drawn → valid.
If the conclusion adds anything not already shown → invalid.

25.5.2 Worked Example

Syllogism Validity by Venn — Worked

Q. Is this valid? Premise 1: All mammals (M) are warm-blooded (P). [A] Premise 2: All whales (S) are mammals (M). [A] Conclusion: All whales (S) are warm-blooded (P). [A]

Step 1: Draw three circles S, P, M. Step 2: Premise 1 — shade the part of M outside P (empty). Step 3: Premise 2 — shade the part of S outside M (empty). Step 4: Reading: S has no region outside M; M has no region outside P. So S has no region outside P. Conclusion: “All whales are warm-blooded” — forced. Valid (Barbara, AAA-1).

25.5.3 Spotting Invalidity by Venn

Venn Test for Invalidity

Premises only universal, conclusion particular → check existential assumption (Boolean reading says invalid).
Premises don’t shade the conclusion region → invalid.
x doesn’t land cleanly → invalid.
Counter-example is constructable → invalid.

25.5.4 Five Tests on the 3-Circle Venn

Useful Identities for 3-Circle Problems

Only A (just A, no B or C) = A − (B ∪ C) = n(A) − n(A∩B) − n(A∩C) + n(A∩B∩C).
Exactly two of A, B, C = sum of pair-intersections − 3 × (A∩B∩C).
Exactly one of A, B, C = total − (pair sums) + 3 × (A∩B∩C) — varies by formula.
At least one = A ∪ B ∪ C.
None of the three = U − (A ∪ B ∪ C).

25.6 Set-Theory Numerical Problems

NTA often phrases set problems with numbers — surveys with overlap.

Set-Theory Numerical — Worked

Q. In a class of 60: 35 like maths, 28 like science, 10 like both. How many like neither?

Step: Like at least one = 35 + 28 − 10 = 53. Like neither = 60 − 53 = 7.

Q. In a survey of 100 people: 60 read newspaper A, 50 read B, 30 read both. How many read only A?

Step: Only A = 60 − 30 = 30. Only B = 50 − 30 = 20. Both = 30. Neither = 100 − (30+20+30) = 20.

Q. Among 200 students: 120 study Hindi, 90 study English, 60 study both. How many study only English?

Step: Only English = 90 − 60 = 30.

25.7 Limitations of the Venn Approach

When Venn Diagrams Get Hard

More than 3 circles — n = 4 needs ellipses (not circles). For n ≥ 5, exact Venn requires special shapes; Euler diagrams are then used.
Existential commitment — Aristotelian reading (all A are B → some A are B) and Boolean reading (no existential commitment) diverge for syllogisms with two universal premises and a particular conclusion.
Probability and uncertainty — Venn is set-theoretic, not probabilistic; for probabilistic relations use Bayes.
Continuous magnitudes — Venn works on categorical class membership, not on degrees.

25.8 Euler vs Venn Diagrams

Euler vs Venn

Euler diagram (Leonhard Euler, 1768) — draws only the actual relations (subset shown by one inside another; disjoint shown by separation). Region absent in actual relation is not drawn.
Venn diagram (John Venn, 1880) — draws all possible intersections, then shades empty ones and marks x in non-empty.

PYQs use the term “Venn” but sometimes mean Euler. For syllogism validity, the Venn version is standard.

25.9 Historical Note

Brief History

Aristotle (4th c. BCE) — categorical logic without diagrams.
Gottfried Wilhelm Leibniz (17th c.) — proposed lines and circles for logic.
Leonhard Euler (1768) — Euler circles.
John Venn (1880) — On the Diagrammatic and Mechanical Representation of Propositions and Reasonings, introducing Venn diagrams.
George Boole (1847) — Boolean algebra; modern reading of Venn diagrams.
Charles Sanders Peirce (1880s onwards) — extensions and existential graphs.

25.10 Practice Questions

Q 01 Origin Easy

The Venn diagram was introduced in 1880 by:

ALeonhard Euler
BJohn Venn
CGeorge Boole
DAristotle

View solution

Correct Option: B

John Venn, 1880. Euler circles were earlier (1768).

Q 02 Proposition Medium

"All teachers are graduates" (proposition A) is best diagrammed as:

ATwo disjoint circles
B"Teachers" entirely inside "Graduates"
CTwo overlapping circles with x in the overlap
DTwo identical circles

View solution

Correct Option: B

A: S ⊆ P → Teachers inside Graduates.

Q 03 Proposition Medium

"Some doctors are women" (proposition I) is best diagrammed as:

ATwo disjoint circles
BOne circle inside the other
CTwo overlapping circles with x in the overlap
DTwo identical circles

View solution

Correct Option: C

I: S ∩ P ≠ ∅ → overlapping circles, with an "x" in the overlap.

Q 04 Proposition Medium

"No fish are mammals" (proposition E) is best diagrammed as:

ATwo disjoint circles
BOne inside the other
CTwo overlapping with x
DIdentical circles

View solution

Correct Option: A

E: S ∩ P = ∅ → disjoint circles.

Q 05 3 Sets Medium

Which diagram best represents the relation among: BIRDS, SPARROWS, CATS?

AAll three overlapping
BSparrows inside Birds; Cats disjoint from Birds
CAll three disjoint
DAll three concentric

View solution

Correct Option: B

All sparrows are birds (S inside B); no cats are birds (C disjoint from B).

Q 06 3 Sets Medium

Which diagram best represents the relation among: TEACHERS, DOCTORS, WOMEN?

AAll three overlapping
BTwo inside one
CAll three disjoint
DDoctors inside Teachers

View solution

Correct Option: A

Some are teachers AND doctors, some are doctors AND women, some are women AND teachers — all three categories partially overlap.

Q 07 3 Sets Medium

Which diagram best represents: PIGEONS, BIRDS, ANIMALS?

AThree concentric circles (Pigeons inside Birds inside Animals)
BAll three disjoint
CAll three overlapping
DPigeons disjoint from Birds

View solution

Correct Option: A

All pigeons are birds; all birds are animals → three concentric circles.

Q 08 Survey Medium

In a class of 50: 30 study English, 25 study Hindi, 10 study both. How many study neither?

View solution

Correct Option: A

Study at least one = 30 + 25 − 10 = 45. Neither = 50 − 45 = 5.

Q 09 Survey Medium

In a survey of 100 households: 60 read newspaper A, 50 read B, 30 read both. How many read ONLY A?

View solution

Correct Option: C

Only A = 60 − 30 = 30.

Q 10 3-set Survey Hard

In a college: 200 like Tea, 150 like Coffee, 100 like Milk. 60 like Tea & Coffee, 50 like Coffee & Milk, 40 like Tea & Milk, 20 like all three. How many like at least one?

A280
B300
C320
D340

View solution

Correct Option: C

n(T∪C∪M) = 200+150+100 − 60−50−40 + 20 = 450 − 150 + 20 = 320.

Q 11 Operations Easy

The notation "A ∩ B" stands for:

AUnion of A and B
BIntersection of A and B
CA is subset of B
DComplement of A in B

View solution

Correct Option: B

∩ = intersection. ∪ = union.

Q 12 De Morgan Hard

By De Morgan's law, (A ∪ B)' equals:

AA' ∪ B'
BA' ∩ B'
CA ∩ B
DA ∪ B

View solution

Correct Option: B

(A ∪ B)' = A' ∩ B'. Dual: (A ∩ B)' = A' ∪ B'.

Q 13 Syllogism Hard

Use a Venn test: "All mammals are warm-blooded. Whales are mammals. Therefore whales are warm-blooded." This syllogism is:

AInvalid — undistributed middle
BInvalid — illicit major
CValid — Barbara (AAA-1)
DInvalid — affirming consequent

View solution

Correct Option: C

Classical valid syllogism, Figure 1, mood AAA = Barbara.

Q 14 Syllogism Hard

"All cats are mammals. Some pets are mammals. Therefore some pets are cats." On a Venn diagram this is:

AValid
BInvalid — undistributed middle
CInvalid — illicit minor
DInvalid — exclusive premises

View solution

Correct Option: B

Middle term "mammals" is the predicate of both an A and an I proposition — undistributed in both. The Venn x for "some pets are mammals" can land outside cats. Undistributed Middle.

Q 15 Euler Hard

Euler diagrams differ from Venn diagrams in that Euler diagrams:

AAlways show all possible intersections
BShow only the actually-existing relations
CUse ellipses instead of circles
DCannot represent the empty set

View solution

Correct Option: B

Euler draws only actual relations; Venn draws all possible intersections and shades empty ones.

Q 16 3 Sets Medium

Which diagram best represents: STUDENTS, BOYS, GIRLS?

ABoys & Girls disjoint; both inside Students
BAll three concentric
CAll three overlapping
DAll three disjoint

View solution

Correct Option: A

Boys and Girls are mutually exclusive but both are subsets of Students.

Q 17 3 Sets Medium

Which diagram best represents: AUTHORS, POETS, NOVELISTS?

APoets and Novelists overlap; both inside Authors
BAll three disjoint
CAll three concentric
DPoets and Novelists disjoint; both inside Authors

View solution

Correct Option: A

Some authors are poets, some novelists; some are both poet and novelist. Both fully inside Authors.

Q 18 Survey Hard

In a survey of 80: 50 like Cricket, 40 like Football, 20 like both. The number who like ONLY ONE sport is:

View solution

Correct Option: B

Only Cricket = 50 − 20 = 30. Only Football = 40 − 20 = 20. Only one = 30 + 20 = 50.

Q 19 Inclusion-Exclusion Medium

If n(A) = 25, n(B) = 30, n(A∩B) = 10, then n(A∪B) =

View solution

Correct Option: C

n(A∪B) = n(A) + n(B) − n(A∩B) = 25 + 30 − 10 = 45.

Q 20 Match Medium

Match each categorical proposition with its Venn picture:

(i)	A — All S are P	(a)	Overlapping; x in overlap
(ii)	E — No S are P	(b)	S inside P
(iii)	I — Some S are P	(c)	Two disjoint circles
(iv)	O — Some S are not P	(d)	Overlapping; x in S only

A(i)-b, (ii)-c, (iii)-a, (iv)-d
B(i)-a, (ii)-b, (iii)-c, (iv)-d
C(i)-c, (ii)-d, (iii)-a, (iv)-b
D(i)-d, (ii)-a, (iii)-b, (iv)-c

View solution

Correct Option: A

A → S inside P; E → disjoint; I → x in overlap; O → x in S − P.

25.11 Quick Recall

Quick recall

Venn diagrams — John Venn 1880. Each circle = a set; overlaps = intersections.
Set vocabulary: ∪ union, ∩ intersection, − difference, ’ complement, ⊆ subset, ∅ empty, U universal, n(A) cardinality.
Inclusion-exclusion (2 sets): n(A∪B) = n(A) + n(B) − n(A∩B).
Inclusion-exclusion (3 sets): n(A∪B∪C) = sum of singles − sum of pairs + triple.
De Morgan: (A∪B)’ = A’∩B’; (A∩B)’ = A’∪B’.
A — All S are P → S inside P (or shade S outside P).
E — No S are P → disjoint circles (or shade S∩P).
I — Some S are P → x in S∩P (overlapping).
O — Some S are not P → x in S − P.
3-circle Venn test for syllogism: shade universal premise first; place x for particular; if conclusion forced → valid.
Word-problem heuristic: “All X are Y” → X inside Y; “Some” → overlap; “No” → disjoint.
Euler vs Venn: Euler shows only actual relations; Venn shows all possible intersections + shades empty ones.
Historical: Aristotle (no diagrams) → Leibniz → Euler 1768 → John Venn 1880 → Boole 1847 → Peirce.
Limitations: > 3 sets needs ellipses; existential commitment differs between Aristotelian and Boolean readings.