Venn Diagrams

Set membership and overlap problems

92 practice questionsPage updated 27 Mar 2026Review metadata pendingPractice questions

Venn diagrams are visual representations used to illustrate relationships between sets. They consist of overlapping circles where each circle represents a set, and the overlapping regions show common elements. Venn diagrams are essential tools for solving syllogism problems, set theory questions, and data interpretation involving multiple categories. In logical reasoning, 2-circle and 3-circle Venn diagrams are most commonly used to analyse relationships between groups and determine valid conclusions.

Key concepts

2-Circle Venn Diagram

A 2-circle Venn diagram represents two sets (A and B) with four distinct regions:

Only A: Elements belonging exclusively to set A
Only B: Elements belonging exclusively to set B
A intersect B (Intersection): Elements common to both sets A and B
Neither A nor B: Elements outside both circles

The universal set (U) contains all elements under consideration. The formula: n(A union B) = n(A) + n(B) - n(A intersect B), where n represents the number of elements.

3-Circle Venn Diagram

A 3-circle Venn diagram represents three sets (A, B, and C) with eight distinct regions:

Only A, Only B, Only C: Elements exclusive to each set
A intersect B only, B intersect C only, A intersect C only: Elements in exactly two sets
A intersect B intersect C (All three): Elements common to all three sets
None of the three: Elements outside all circles

Key formula: n(A union B union C) = n(A) + n(B) + n(C) - n(A intersect B) - n(B intersect C) - n(A intersect C) + n(A intersect B intersect C)

Intersection and Union

Understanding intersection and union is crucial for Venn diagram problems:

Intersection (A intersect B): The region where circles overlap, representing elements present in BOTH sets. 'Some A are B' statements refer to this region.

Union (A union B): The entire area covered by both circles combined, representing elements in A OR B or both. 'All A are B' means set A is entirely within set B.

Complement (A'): Everything outside set A within the universal set.

Set Relationships and Syllogisms

Venn diagrams help visualise syllogistic statements:

'All A are B' -> Circle A is completely inside circle B
'No A are B' -> Circles A and B do not overlap at all
'Some A are B' -> Circles A and B overlap (intersection exists)
'Some A are not B' -> Part of circle A exists outside circle B

For valid syllogistic conclusions, the Venn diagram must support the conclusion in ALL possible configurations that satisfy the premises.

Worked examples

Example 1

In a survey of 100 people: 60 like tea, 50 like coffee, and 30 like both. How many like neither?

Solution

Given:
Total = 100
n(Tea) = 60
n(Coffee) = 50
n(Tea intersect Coffee) = 30

Step 1: Find those who like only tea
Only Tea = 60 - 30 = 30

Step 2: Find those who like only coffee
Only Coffee = 50 - 30 = 20

Step 3: Calculate union (at least one beverage)
n(Tea union Coffee) = Only Tea + Only Coffee + Both
= 30 + 20 + 30 = 80

Step 4: Find those who like neither
Neither = Total - n(Tea union Coffee)
Neither = 100 - 80 = 20

Answer: 20 people like neither tea nor coffee.

Example 2

In a class of 80 students: 45 study Maths, 40 study Physics, 35 study Chemistry. 20 study Maths and Physics, 18 study Physics and Chemistry, 22 study Maths and Chemistry, and 10 study all three. How many study exactly one subject?

Solution

Given 3-circle Venn diagram problem.

Step 1: Find students in exactly two subjects
Maths and Physics only = 20 - 10 = 10
Physics and Chemistry only = 18 - 10 = 8
Maths and Chemistry only = 22 - 10 = 12

Step 2: Find students in exactly one subject
Only Maths = 45 - 10 - 10 - 12 = 13
Only Physics = 40 - 10 - 10 - 8 = 12
Only Chemistry = 35 - 12 - 10 - 8 = 5

Step 3: Calculate exactly one subject
Exactly one = 13 + 12 + 5 = 30

Verification: Total accounted = 13+12+5+10+8+12+10 = 70
Students studying none = 80 - 70 = 10

Answer: 30 students study exactly one subject.

Example 3

In a group: All apples are fruits. Some fruits are red. No vegetable is red. Which conclusion follows: (1) Some apples are red, (2) No vegetable is a fruit?

Solution

Analysing the premises with Venn diagrams:

Premise 1: 'All apples are fruits' -> Apple circle inside Fruit circle
Premise 2: 'Some fruits are red' -> Fruit and Red circles overlap
Premise 3: 'No vegetable is red' -> Vegetable and Red circles don't touch

Checking Conclusion 1: 'Some apples are red'
Since Apple is inside Fruit, and Fruit overlaps with Red, Apple MAY or MAY NOT overlap with Red. The overlap area between Fruit and Red could be entirely outside Apple. Therefore, this conclusion does NOT necessarily follow.

Checking Conclusion 2: 'No vegetable is a fruit'
Vegetable doesn't intersect with Red, but Fruit may extend beyond Red. Vegetable and Fruit circles could still overlap (in the non-red region of Fruit). Therefore, this conclusion does NOT necessarily follow.

Answer: Neither conclusion follows necessarily.

Example 4

In a company, 120 employees know Java, 100 know Python, 80 know C++. 60 know Java and Python, 40 know Python and C++, 50 know Java and C++, and 25 know all three languages. How many know at least one language?

Solution

Using the 3-set union formula:

Given:
n(Java) = 120
n(Python) = 100
n(C++) = 80
n(Java intersect Python) = 60
n(Python intersect C++) = 40
n(Java intersect C++) = 50
n(Java intersect Python intersect C++) = 25

Formula: n(A union B union C) = n(A) + n(B) + n(C) - n(A intersect B) - n(B intersect C) - n(A intersect C) + n(A intersect B intersect C)

Substituting:
= 120 + 100 + 80 - 60 - 40 - 50 + 25
= 300 - 150 + 25
= 175

Answer: 175 employees know at least one programming language.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

In a Venn diagram with two overlapping circles A and B, which region represents elements that are in A only?

The overlapping region
The region of A that does not overlap with B
The region of B that does not overlap with A
The area outside both circles

Answer: B. The region of A that does not overlap with B

Explanation

In a two-circle Venn diagram:
- Only A = The part of circle A that does NOT overlap with circle B
- Only B = The part of circle B that does NOT overlap with circle A
- A intersect B = The overlapping region
- Outside = Elements in neither A nor B

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

In a Venn diagram with two circles A and B, what does the intersection (shaded overlapping region) represent?

Elements in A only
Elements in B only
Elements in both A and B
Elements in neither A nor B

Answer: C. Elements in both A and B

Explanation

The intersection (A intersect B) represents the overlapping region of the two circles.
This region contains elements that belong to BOTH sets A and B simultaneously.
It is commonly shaded in Venn diagram problems to highlight common elements.

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

If circle A has 40 elements, circle B has 35 elements, and their intersection has 15 elements, how many elements are in A only?

Answer: B. 25

Explanation

Elements in A only = Total in A - Elements in intersection
= 40 - 15
= 25
The intersection (15 elements) must be subtracted from circle A's total to get only-A elements.

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Venn Diagrams

Key concepts

2-Circle Venn Diagram

3-Circle Venn Diagram

Intersection and Union

Set Relationships and Syllogisms

Data Interpretation with Venn Diagrams

Special Venn Diagram Cases

Worked examples

Example 1

Example 2

Example 3

Example 4

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Key concepts

2-Circle Venn Diagram

3-Circle Venn Diagram

Intersection and Union

Set Relationships and Syllogisms

Data Interpretation with Venn Diagrams

Special Venn Diagram Cases

Worked examples

Example 1

Example 2

Example 3

Example 4

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

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