Venn Diagram Calculator
Calculate set relationships, unions, intersections, and complements using Venn diagrams
Configure Your Venn Diagram
Number of Sets
Total number of elements in the universal set
Number of elements in set A
Number of elements in set B
Known Set Relationship
Venn Diagram Results
Set Operations
Complements
Formulas Used
|A ∪ B| = |A| + |B| - |A ∩ B| (Inclusion-Exclusion Principle)
|A \ B| = |A| - |A ∩ B| (Set Difference)
|A Δ B| = |A ∪ B| - |A ∩ B| (Symmetric Difference)
Set Theory Notation
Intersection
Elements in both sets
Union
Elements in either set
Difference
Elements in first set only
Symmetric Difference
Elements in exactly one set
Complement
Elements not in the set
Venn Diagram Regions
Two-Set Venn Diagram
Quick Tips
Union includes all elements from both sets
Intersection includes only shared elements
Check that set sizes don't exceed the universal set
Intersection cannot be larger than either individual set
Worked Example
Problem Setup
Survey of 50 students about subject preferences:
- • 20 students like Mathematics (Set A)
- • 20 students like Science (Set B)
- • 5 students like both subjects
- • How many like exactly one subject?
Given Information:
- |U| = 50 (total students)
- |A| = 20 (like math)
- |B| = 20 (like science)
- |A ∩ B| = 5 (like both)
Solution
Calculations
1. Only Math: |A \ B| = |A| - |A ∩ B| = 20 - 5 = 15
2. Only Science: |B \ A| = |B| - |A ∩ B| = 20 - 5 = 15
3. Union: |A ∪ B| = |A| + |B| - |A ∩ B| = 20 + 20 - 5 = 35
4. Neither: |U| - |A ∪ B| = 50 - 35 = 15
Answer: 30 students like exactly one subject (15 + 15)
Understanding Venn Diagrams
What are Venn Diagrams?
Venn diagrams are visual representations used in set theory to show logical relationships between different sets. They use overlapping shapes (usually circles) to illustrate how sets intersect, unite, or differ from each other.
Key Concepts
- •Universal Set (U): Contains all possible elements
- •Intersection (∩): Elements common to multiple sets
- •Union (∪): All elements in any of the sets
- •Complement ('): Elements not in the set
Key Formulas
Inclusion-Exclusion Principle
For calculating union from individual sets
Set Difference
Elements in A but not in B
Symmetric Difference
Elements in exactly one set
Applications
Probability
Calculate probabilities of combined events
Logic & Computer Science
Boolean operations and database queries
Survey Analysis
Analyze overlapping responses and preferences