Free · Probability and statistical models

Two and Three Set Count Relationships

Solve unions, intersections, exclusive regions and complements for two or three sets.

  • Formula & worked example
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

Within a universe of 100, sets of 40 and 50 with overlap 10 have union 80 and exclusive regions 30 and 40.

|A∪B|=|A|+|B|−|A∩B|; complements subtract from the universe

Worked example

Enter these known values and leave the other values blank.

|U|
100
|A|
40
|B|
50
|A∩B|
10
|A\B|
30
|B\A|
40
|A∪B|
80
|AΔB|
70
|A'|
60
|B'|
50
|(A\B)'|
70
|(B\A)'|
60
|(A∪B)'|
20
|(A∩B)'|
90
|(AΔB)'|
30

Assumptions and limitations

  • Counts are whole nonnegative values in a common finite universe.
  • Intersections cannot exceed their component set counts; a union cannot exceed the universe.
  • In three-set mode, a pairwise intersection includes the triple intersection.

Common questions

Which assumptions govern this model?

Counts are whole nonnegative values in a common finite universe.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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