Calculate Venn diagram regions for 2 or 3 sets. Find unions, intersections, complements, conditional probabilities, Jaccard similarity, and all exclusive regions with visual diagrams.
The Venn diagram calculator computes all region counts and probabilities for two or three overlapping sets inside a universal set. Enter the set sizes and intersections to get the exclusive regions, conditional probabilities, similarity metrics, and the full region breakdown.
For two sets, it calculates the union, intersection, symmetric difference, and several overlap metrics such as Jaccard and Dice. For three sets, it splits the diagram into all eight regions so you can see exactly how many elements fall into each combination.
That makes it useful for survey analysis, set-relationship exercises, and probability problems where overlap between events is the main thing you need to understand.
Venn diagrams are a compact way to see overlap, exclusion, and shared membership at the same time. This calculator keeps the bookkeeping out of the way so you can focus on what the regions mean.
It is most helpful when you need to decompose a problem into exclusive regions, compare set similarity, or translate set counts into probabilities.
|A∪B| = |A| + |B| − |A∩B|. A only = |A| − |A∩B|. Neither = |U| − |A∪B|. For three sets: |A∪B∪C| = |A| + |B| + |C| − |A∩B| − |A∩C| − |B∩C| + |A∩B∩C|.
Result: Union = 75, A only = 35, B only = 15, Neither = 25
|A∪B| = 60 + 40 − 25 = 75. A only = 60 − 25 = 35 (in A but not B). B only = 40 − 25 = 15. Neither = 100 − 75 = 25. Jaccard similarity = 25/75 = 0.333 (moderate overlap).
In modern data science, Venn diagrams help visualize overlap between data clusters, feature sets, or model predictions. The Jaccard index is used as a loss function in image segmentation, the Dice coefficient in medical image analysis, and the overlap coefficient in gene set enrichment. Understanding these metrics starts with the Venn diagram framework.
Venn diagrams for 4+ sets exist but become complex — a 4-set diagram requires ellipses or non-circular shapes, and has 16 regions. For 5+ sets, the diagrams become impractical. This is why most practical applications use 2 or 3 sets and rely on tables or UpSet plots for higher dimensions.
De Morgan's laws connect complements of unions and intersections: (A∪B)ᶜ = Aᶜ∩Bᶜ and (A∩B)ᶜ = Aᶜ∪Bᶜ. On a Venn diagram, the complement of the union is the "neither" region, while the complement of the intersection is everything except the overlap. These laws extend to three or more sets.
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Union (A∪B) includes everything in A or B or both — it's the "or" operation. Intersection (A∩B) includes only elements in both A and B — it's the "and" operation. The Venn diagram shows union as both circles combined, intersection as only the overlap.
The symmetric difference (A△B) contains elements in A or B but not both — it's the exclusive or (XOR) of sets. It equals |A only| + |B only| = |A∪B| − |A∩B|. Useful for measuring how different two sets are.
Jaccard = |A∩B| / |A∪B|. It ranges from 0 (no overlap) to 1 (identical sets). Above 0.5 indicates strong overlap. It's used in machine learning to compare predicted vs actual labels, in ecology to compare species sets, and in search engines for document similarity.
Lift compares observed intersection to what you'd expect if the sets were independent: Lift = (|A∩B| × |U|) / (|A| × |B|). Lift = 1 means no association. Lift > 1 means positive association (they co-occur more than expected). Common in market basket analysis.
Yes! If U represents equally likely outcomes, each region count divided by |U| gives the probability. P(A∪B), P(A∩B), P(A|B) are directly calculated. This is the counting method for discrete probability.
The universal set determines the "neither" region and allows probability calculations. Without it, you can't determine P(A) or the complement. In surveys, U is the total number of respondents.