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 2 or 3 overlapping sets within a universal set. Enter set sizes and intersection sizes to find exclusive regions, conditional probabilities, similarity metrics, and a complete breakdown of every Venn diagram region.
For two sets, it calculates union, intersection, symmetric difference, and four similarity metrics (Jaccard, Dice, overlap coefficient, lift). For three sets, it decomposes the diagram into all 8 distinct regions, showing exactly how many elements fall in each combination of sets.
This tool is essential for visualizing survey data, understanding set relationships in combinatorics problems, analyzing feature overlap in data science, and solving probability problems involving multiple events. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case. Use the example pattern when troubleshooting unexpected results. Validate that outputs match your chosen standards.
Venn diagrams are one of the most intuitive ways to understand set relationships and probability. This calculator does the arithmetic for you — enter set sizes and intersections to get every region count, probability, conditional probability, and similarity metric.
Perfect for survey analysis, combinatorics homework, data science feature overlap, and any scenario where you need to decompose overlapping groups into exclusive regions.
|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.
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.