Venn Diagram Calculator

Enter the sizes of your sets A, B, and optionally C, along with their intersections, and the Venn Diagram Calculator computes every region — union, intersection, exclusive parts, and symmetric difference. Choose between a 2-set or 3-set diagram, plug in your cardinalities, and get a full breakdown of all overlapping and non-overlapping regions instantly.

Total number of elements in the universe (optional).

Total number of elements in Set A.

Total number of elements in Set B.

Total number of elements in Set C (3-set diagrams only).

Number of elements shared between Set A and Set B.

Number of elements shared between Set A and Set C (3-set only).

Number of elements shared between Set B and Set C (3-set only).

Number of elements shared by all three sets (3-set only).

Results

Union |A ∪ B|

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A Only (exclusive)

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B Only (exclusive)

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C Only (exclusive)

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A ∩ B only (not C)

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A ∩ C only (not B)

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B ∩ C only (not A)

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A ∩ B ∩ C (all three)

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Symmetric Difference

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Neither (outside all sets)

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Venn Diagram Region Breakdown

Results Table

Frequently Asked Questions

What is a Venn diagram?

A Venn diagram is a visual representation of sets and their relationships, using overlapping circles. Each circle represents a set, and the overlapping areas show elements that belong to more than one set simultaneously. They are widely used in logic, statistics, probability, and everyday problem-solving.

What is the inclusion-exclusion principle?

The inclusion-exclusion principle is a counting technique used to find the size of the union of sets. For two sets: |A ∪ B| = |A| + |B| − |A ∩ B|. For three sets, you add all three set sizes, subtract pairwise intersections, then add back the triple intersection: |A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |A ∩ C| − |B ∩ C| + |A ∩ B ∩ C|.

How do I calculate the intersection between three sets?

To find the region belonging exclusively to all three sets (A ∩ B ∩ C), you need the total count of elements that appear in Set A, Set B, and Set C simultaneously. This value is entered directly into the calculator. From it, each pairwise exclusive region (e.g., A ∩ B but not C) is derived by subtracting the triple intersection from the pairwise intersection.

What is the union if |A| = 10, |B| = 12, and |A ∩ B| = 4?

Using the inclusion-exclusion principle: |A ∪ B| = |A| + |B| − |A ∩ B| = 10 + 12 − 4 = 18. So there are 18 unique elements across both sets combined.

What is the symmetric difference of two sets?

The symmetric difference of two sets A and B contains all elements that belong to either A or B, but not both. It is calculated as |A| + |B| − 2×|A ∩ B|. For example, if |A| = 10, |B| = 12, and |A ∩ B| = 4, the symmetric difference is 10 + 12 − 8 = 14.

What does 'A only' mean in the Venn diagram results?

'A only' refers to elements that are exclusively in Set A and do not appear in any other set. It is calculated as |A| − |A ∩ B| for a 2-set diagram. For a 3-set diagram, it is |A| − |A ∩ B| − |A ∩ C| + |A ∩ B ∩ C|.

Can this calculator handle 3-set Venn diagrams?

Yes. Select '3 Sets' at the top, then enter the sizes of Sets A, B, and C along with all pairwise intersections (A ∩ B, A ∩ C, B ∩ C) and the triple intersection (A ∩ B ∩ C). The calculator will compute every region in the 3-circle diagram, including exclusive zones and the full union.

What is the 'neither' region in a Venn diagram?

The 'neither' region represents elements in the universal set that do not belong to any of the defined sets. It is calculated as |U| − |A ∪ B| (or |U| − |A ∪ B ∪ C| for three sets). If no universal set is provided, this value will be shown as not applicable.

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