Set Operations Explorer: 3D Venn Diagrams

Set Sizes

|A| 24

|B| 24

Overlap fraction 40%

Operation

Live Counts

|A|

|B|

|A ∩ B|

|A ∪ B|

|A ∪ B| = |A| + |B| − |A ∩ B|

Selected result set

Excluded elements

How it works

Each glowing point is one element. Set A lives in the left sphere, set B in the right sphere; the lens-shaped overlap region holds elements that belong to both sets simultaneously — that's A ∩ B. Every other point belongs to exactly one set.

A ∪ B = { x : x ∈ A or x ∈ B }
A ∩ B = { x : x ∈ A and x ∈ B }
A ∖ B = { x : x ∈ A and x ∉ B }
B ∖ A = { x : x ∈ B and x ∉ A }
A △ B = (A ∖ B) ∪ (B ∖ A)

Inclusion–exclusion:
|A ∪ B| = |A| + |B| − |A ∩ B|

|A|, |B| sliders — the total element count of each set.
Overlap fraction — what share of the smaller set's elements also belong to the other set (drives both the geometric overlap and the exact |A ∩ B| count).
Operation buttons — pick which resulting set lights up in accent color; every other point fades to gray. The formula box and counts update to match the selected operation exactly.

This is the same combinatorial machinery behind SQL joins (INNER/LEFT/FULL), Venn-diagram probability, and the inclusion–exclusion principle used to count unions of overlapping events.

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