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What Is a Venn Diagram?
A Venn diagram uses overlapping circles to show relationships between sets. Each circle represents one set, the overlapping portions show elements shared between sets (intersections), and the area inside all circles combined represents the union. British logician John Venn introduced the diagrams in 1880 as a way to visualize logical propositions and set theory.
The term "Venn diagram" is sometimes used loosely to describe what is technically an Euler diagram (see below), but in mathematics education a Venn diagram always draws every possible overlapping region, even when some regions contain no elements. This makes the complete logical space visible at a glance.
How to Read the Regions
| Region | Notation | Meaning | Example |
|---|---|---|---|
| A only | A − B | Elements in A but not in B | Students who study only Maths |
| B only | B − A | Elements in B but not in A | Students who study only Statistics |
| Intersection | A ∩ B | Elements in both A and B | Students who study both |
| Union | A ∪ B | All elements in A or B or both | All students who study either subject |
| Triple intersection | A ∩ B ∩ C | Elements in all three sets | Students who study all three subjects |
The Two Key Formulas
The intersection is subtracted because elements in both A and B are counted once in |A| and once in |B|, so they would otherwise appear twice. Subtracting |A ∩ B| corrects this double-counting.
The triple intersection is added back because subtracting all three pairwise intersections removes the triple-overlap region three times, yet it was only double-counted three times, so it must be restored once.
Where Venn Diagrams Get Used
Venn Diagram vs Euler Diagram
A Venn diagram always draws every possible overlapping region, even if that region is empty. An Euler diagram only draws regions that actually contain elements, so disjoint sets appear as separate circles with no overlap. Both are valid; Euler diagrams are often cleaner for sparse data, but Venn diagrams are preferred in formal mathematics and education because they make the complete logical space explicit.
Common Mistakes
The most frequent error is placing an element that belongs to all three sets into just the pairwise intersection, rather than the innermost triple-overlap region. Each element must appear in exactly one region: the most specific region it belongs to. Another common error is double-counting the intersection when calculating a union by hand. The formula |A ∪ B| = |A| + |B| − |A ∩ B| exists precisely to prevent this.
Related Topics
Sources and further reading:
- Venn, J. (1880). On the Diagrammatic and Mechanical Representation of Propositions and Reasonings. Philosophical Magazine, 9(59), 1–18.
- Weisstein, E. W. Venn Diagram. MathWorld
- Khan Academy — Probability Library
Frequently Asked Questions
A Venn diagram uses overlapping circles to show the relationships between sets. Each circle represents a set of elements. The region where circles overlap contains elements that belong to both sets (the intersection). The area covered by all circles combined represents the union. Introduced by John Venn in 1880, they are now standard in mathematics, statistics, probability, and logic.
The intersection (A ∩ B) contains only elements that appear in both A and B. The union (A ∪ B) contains every element that appears in A or B or both. In a Venn diagram, the intersection is the overlapping region, and the union is the total area covered by all circles combined.
Use Element mode when you have a list of items and want to see which sets they belong to. Type the elements into each set, and the tool calculates every intersection, union, and difference automatically. Use Region mode when you already know the count for each region (common in probability exam questions) and just need the diagram rendered and totals confirmed. Both modes update the same SVG diagram.
The formula is: |A ∪ B| = |A| + |B| − |A ∩ B|. The intersection must be subtracted because elements in both sets are counted once in |A| and once in |B|, so they would otherwise be counted twice. If A = {1, 2, 3} and B = {3, 4, 5}, then |A ∪ B| = 3 + 3 − 1 = 5, which matches {1, 2, 3, 4, 5}.
Select "3 Sets" from the dropdown in either Element or Region mode. A third input block or set of region fields will appear. The diagram will show all seven regions: A only, B only, C only, A∩B (excluding C), A∩C (excluding B), B∩C (excluding A), and the central triple intersection A∩B∩C. The union formula for three sets is |A ∪ B ∪ C| = |A| + |B| + |C| − |A∩B| − |A∩C| − |B∩C| + |A∩B∩C|.
Yes. Click "Download SVG" to save a vector file. SVG files scale to any size without quality loss, so they are suitable for presentations, printed reports, and academic assignments. You can also click "Copy SVG" to paste the diagram code directly into another application, or "Print" to send it to a printer.
A Venn diagram always shows every possible overlapping region between sets, even if some contain no elements. An Euler diagram only draws regions that actually contain elements, so disjoint sets appear as separate circles. Euler diagrams are often cleaner for sparse relationships, but Venn diagrams are the standard in mathematics education because they display the complete logical space explicitly.
No. All calculations and diagram rendering happen entirely in your browser using JavaScript. No element data or values are sent to any server. You can use this tool with sensitive or personal data without privacy concerns.