Venn Diagram Fundamentals

Venn diagram questions appear in approximately 3–5 of the 29 Decision Making questions. They test your ability to interpret overlapping sets, read information from diagrams, and draw logical conclusions. This lesson covers the foundations: how to read and construct 2-circle and 3-circle Venn diagrams, what each region represents, and how to extract information accurately.

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What Is a Venn Diagram?

A Venn diagram uses overlapping circles within a rectangle to show the relationships between sets (groups). The rectangle represents the universal set (everything under consideration), and each circle represents a specific subset.

Key Terminology

Term	Meaning
Universal set (U)	Everything under consideration (the rectangle)
Set	A defined group (represented by a circle)
Intersection (A ∩ B)	Members belonging to BOTH A and B
Union (A ∪ B)	Members belonging to A OR B OR both
Complement (A')	Members NOT in A (everything in the rectangle outside A)
Only A	Members in A but not in any other set

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Two-Circle Venn Diagrams

A two-circle Venn diagram creates four distinct regions:

Region	Description	Notation
1	In A only (not in B)	A ∩ B'
2	In both A and B	A ∩ B
3	In B only (not in A)	A' ∩ B
4	In neither A nor B	A' ∩ B' (outside both circles)

Reading Numbers from a Two-Circle Diagram

If a question gives you numbers in each region, you can calculate:

Quantity	How to calculate
Total in A	Region 1 + Region 2
Total in B	Region 2 + Region 3
Total in A and B (intersection)	Region 2
Total in A or B (union)	Region 1 + Region 2 + Region 3
Total in neither	Region 4
Grand total	Region 1 + Region 2 + Region 3 + Region 4

Worked Example: Two Circles

In a group of 80 medical students, 55 study anatomy, 40 study physiology, and 25 study both anatomy and physiology. How many students study neither subject?

Step 1: Identify the regions.

• Both anatomy and physiology (intersection) = 25
• Anatomy only = 55 − 25 = 30
• Physiology only = 40 − 25 = 15
• Neither = 80 − (30 + 25 + 15) = 80 − 70 = 10

Region	Count
Anatomy only	30
Both	25
Physiology only	15
Neither	10
Total	80

Answer: 10 students study neither subject.

Common Error: A frequent mistake is adding 55 + 40 = 95 and then subtracting from 80 to get −15, which is impossible. This happens when you forget that the 25 students in both subjects are counted in BOTH the 55 and the 40. Always subtract the intersection first.

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Three-Circle Venn Diagrams

A three-circle Venn diagram creates eight distinct regions:

Region	Description
1	In A only
2	In A and B only (not C)
3	In B only
4	In A and C only (not B)
5	In all three (A, B, and C)
6	In B and C only (not A)
7	In C only
8	In none (outside all circles)

Working with Three Circles

The key principle is to work from the inside out:

1. Start with the innermost region — the intersection of all three sets.
2. Then fill in the two-set intersections — subtract the "all three" value.
3. Then fill in the "only" regions — subtract all overlapping parts.
4. Finally, calculate the "none" region — subtract the total of all other regions from the universal set.

Worked Example: Three Circles