Venn Diagrams questions with answers and concepts
Learn venn diagrams with concept notes and worked examples, then practise focused questions with explained answers. Build accuracy for placement and competitive aptitude tests before trying timed mode.
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.
Data Interpretation with Venn Diagrams
When numerical data is given with overlapping categories:
1. Start from the innermost region (all sets intersection) and work outward
2. Calculate 'only' regions by subtracting overlapping counts from total
3. Use variables (x, y, z) for unknown regions when setting up equations
4. Sum of all regions equals the total number of elements
Common applications: students studying multiple subjects, employees with multiple skills, survey respondents selecting multiple options.
Special Venn Diagram Cases
Be aware of special configurations:
Disjoint sets: Circles that don't touch (no common elements)
Subset relationship: One circle completely inside another
Equal sets: Two circles perfectly overlapping (all elements common)
For maximum and minimum value problems:
Maximum overlap occurs when the smaller set is entirely within the larger
Minimum overlap: n(A) + n(B) - n(U) when dealing with two sets within a universal set
Use the principle: n(A intersect B) >= n(A) + n(B) - n(U) for minimum intersection
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(Coffee) = 50
n(Tea intersect Coffee) = 30
Step 1: Find those who like only tea
Only
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
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
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(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:
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 class of 200 students, 80 passed in Math, 70 passed in Science, and 30 passed in both. How many failed in both subjects?
Answer: B. 80
ExplanationStep 1: Students who passed at least one subject
n(Math union Science) = 80 + 70 - 30 = 120
Step 2: Students who failed both = Total - Passed at least one
= 80 students
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 3-circle Venn diagram with A, B, C: if n(A only) = 20, n(B only) = 15, n(C only) = 10, n(A intersect B only) = 8, n(A intersect C only) = 6, n(B intersect C only) = 5, and n(A intersect B intersect C) = 3, what is n(A union B union C)?
Answer: A. 67
Explanationn(A union B union C) = Sum of all distinct regions
This includes: only A, only B, only C, pairwise intersections only, and the triple intersection.
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
In a Venn diagram with three sets A, B, and C, what is the formula for n(A intersect B intersect C')?
Answer: A. Elements in A and B but not in C
ExplanationA intersect B intersect C' (or A intersect B - C) represents elements that are:
- In set A
- In set B
- NOT in set C
This is the region where A and B overlap, excluding the part that intersects with C.
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
Common mistakes and useful habits
- Always start drawing from the innermost intersection (all sets overlap) and work your way outward to avoid double-counting errors
- When given 'Only A' values directly, you don't need to subtract; but when given total for set A, subtract all overlapping regions to get 'Only A'
- For syllogism questions, remember that 'Some A are B' allows for the possibility that 'All A are B' - it's a weaker claim
- Use variables for unknown regions when the problem requires setting up equations - this makes solving systematic
- In maximum-minimum problems, the maximum intersection equals the size of the smaller set, and minimum intersection follows: Max(0, n(A) + n(B) - Total)
- When a conclusion states definitively that something IS or IS NOT true, check if it holds in ALL valid Venn configurations that satisfy the premises
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