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Venn Diagram Calculator

Free Venn diagram calculator. Set operations with 2 or 3 sets. Union, intersection, complement, symm

Run CalculatorExplore data analysis and statistical calculations

Why This Statistical Analysis Matters

Why: Statistical calculator for analysis.

How: Enter inputs and compute results.

⊂
STATISTICSSet Theory

Interactive Venn Diagram — 2 & 3 Sets

Enter set sizes or probabilities. Get unions, intersections, complements, and region counts. Hover over regions to highlight.

Joint Probability →Probability →

Real-World Scenarios — Click to Load

Total population
|A∪B|
155
|A'|
80
|B'|
120
|A△B|
110
venn_diagram.sh
CALCULATED
$ compute_venn --sets=2 --mode="counts"
|A∪B|
155
|A△B|
110
|A'|
80
|B'|
120
Share:
Venn Diagram Result
|A∪B|
155
numbervibe.com/calculators/statistics/venn-diagram-calculator

Venn Diagram

75A only35B only45A∩B45NeitherAB

Region Values

Calculation Breakdown

UNION
|A∪B| = |A| + |B| − |A∩B|
155
ext{Inclusion}- ext{exclusion}
|A only| = |A| − |A∩B|
75
|B only| = |B| − |A∩B|
35
SYMMETRIC DIFF
|A△B| = |A| + |B| − 2|A∩B|
110
ext{Symmetric} ext{difference}

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

  • • Inclusion-exclusion: |A ∪ B| = |A| + |B| − |A ∩ B|
  • • For 3 sets: |A∪B∪C| = |A|+|B|+|C| − |A∩B| − |A∩C| − |B∩C| + |A∩B∩C|
  • • De Morgan: (A∪B)' = A'∩B' and (A∩B)' = A'∪B'
  • • Symmetric difference: A △ B = (A−B)∪(B−A) = |A|+|B|−2|A∩B|
  • • Complement: |A'| = |U| − |A|

Did You Know?

📐John Venn invented the diagram in 1880 to illustrate logical relations. Euler diagrams are similar but may not show all intersections.Source: Wikipedia
🎯Venn diagrams are used in survey analysis, logic, probability, set theory, and database query design.Source: Khan Academy
📊For 2 sets there are 4 regions; for 3 sets there are 8 regions. A Venn diagram for n sets has 2^n regions.Source: Wolfram MathWorld
🔬In bioinformatics, Venn diagrams visualize gene overlap across tissues or conditions.Source: NIST
🗳️Election analysts use overlapping sets to model voter preferences across multiple parties.Source: OpenIntro
🎓Language learners often learn in overlapping regions: Spanish, French, and German students.Source: MIT OCW

Worked Example (2 Sets)

Example: 120 students take Math, 80 take Science, 45 take both. Total: 200.

Math only: 120 − 45 = 75. Science only: 80 − 45 = 35.

Union: 120 + 80 − 45 = 155. Neither: 200 − 155 = 45.

How the Math Works

1. Inclusion-Exclusion (2 sets)

|A ∪ B| = |A| + |B| − |A ∩ B|. We subtract the intersection because it is counted twice.

2. Inclusion-Exclusion (3 sets)

|A∪B∪C| = |A|+|B|+|C| − |A∩B| − |A∩C| − |B∩C| + |A∩B∩C|. The triple intersection is added back.

3. Region counts

A only = |A| − |A∩B|. B only = |B| − |A∩B|. Neither = |U| − |A∪B|.

4. De Morgan's Laws

Complement of union: (A∪B)' = A'∩B'. Complement of intersection: (A∩B)' = A'∪B'.

5. Symmetric difference

A △ B = elements in A or B but not both. |A△B| = |A| + |B| − 2|A∩B|.

Common Use Cases

Survey Analysis

Overlap of respondents (e.g., Math vs Science students).

Employee Skills

Skills overlap (Python, SQL, etc.).

Disease Screening

Sensitivity/specificity, true positives vs false positives.

Voter Demographics

Party preference overlap across regions.

App Usage

Overlap of users across multiple apps.

Gene Expression

Overlap of genes across tissues or conditions.

Expert Tips

Consistency

Ensure |A∩B| ≤ min(|A|,|B|). For 3 sets, check all pairwise intersections.

Universal set

Use |U| when you know the total population; otherwise leave 0 for probability mode.

Probabilities

Use 0–1 for P(A), P(B), P(A∩B). P(A∪B) = P(A) + P(B) − P(A∩B).

Visualization

Hover over regions to highlight. The bar chart shows all region counts.

Formulas at a Glance

|A∪B| = |A|+|B|−|A∩B|
2-set union
|A'| = |U|−|A|
Complement
|A△B| = |A|+|B|−2|A∩B|
Symmetric diff
(A∪B)' = A'∩B'
De Morgan

Frequently Asked Questions

When should I use counts vs probabilities?

Use counts when you have survey data (e.g., 120 students). Use probabilities when you have P(A), P(B), P(A∩B) as fractions (0–1).

When is |A∩B| invalid?

|A∩B| cannot exceed min(|A|,|B|). For example, if 50 people like A and 30 like B, at most 30 can like both.

What is the symmetric difference?

A △ B = elements in A or B but not both. Useful for "exclusive or" logic.

How do I interpret the 3-set diagram?

Eight regions: A only, B only, C only, A∩B only, A∩C only, B∩C only, A∩B∩C, and none (outside all).

Disclaimer: Venn diagrams are for educational and analytical visualization. Ensure inputs satisfy |A∩B| ≤ min(|A|,|B|) and |A∪B| ≤ |U| when U is known.

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